February 2017 arXiv:yymm.nnnnn [hep-th]D(13.3): standard

Dynamics of D-branes II. The standard action

— an analogue of the Polyakov action for (fundamental, stacked) D-branes

Chien-Hao Liu and Shing-Tung Yau

Abstract

We introduce a new action S(ρ,h;Φ,g,B,C)standard for D-branes that is to D-branes as the Polyakov

action is to fundamental strings. This ‘standard action’ is abstractly a non-Abelian gaugedsigma model — based on maps ϕ : (XAz, E;∇) → Y from an Azumaya/matrix manifoldXAz with a fundamental module E with a connection ∇ to Y — enhanced by the dila-ton term, the gauge-theory term, and the Chern-Simons/Wess-Zumino term that couples(ϕ,∇) to Ramond-Ramond field. In a special situation, this new theory merges the theoryof harmonic maps and a gauge theory, with a nilpotent type fuzzy extension. With theanalysis developed in D(13.1) (arXiv:1606.08529 [hep-th]) for such maps and an improvedunderstanding of the hierarchy of various admissible conditions on the pairs (ϕ,∇) beyondD(13.2.1) (arXiv:1611.09439 [hep-th]) and how they resolve the built-in obstruction to pull-push of covariant tensors under a map from a noncommutative manifold to a commutativemanifold, we develop further in this note some covariant differential calculus needed andapply them to work out the first variation — and hence the corresponding equations of mo-tion for D-branes — of the standard action and the second variation of the kinetic term formaps and the dilaton term in this action. Compared with the non-Abelian Dirac-Born-Infeldaction constructed in D(13.1) along the same line, the current note brings the Nambu-Goto-string-to-Polyakov-string analogue to D-branes. The current bosonic setting is the first steptoward the dynamics of fermionic D-branes (cf. D(11.2): arXiv:1412.0771 [hep-th]) and theirquantization as fundamental dynamical objects, in parallel to what happened to the theoryof fundamental strings during years 1976–1981.

Key words: D-brane; admissible condition; standard action, enhanced non-Abelian gauged sigma model;Azumaya manifold, C∞-scheme, harmonic map; first and second variation, equations of motion

MSC number 2010: 81T30, 35J20; 16S50, 14A22, 35R01

Acknowledgements. We thank Andrew Strominger, Cumrun Vafa for influence to our understanding of strings, branes,and gravity. C.-H.L. thanks in addition Pei-Ming Ho for a discussion on Ramond-Ramond fields and literature guide

and Chenglong Yu for a discussion on admissible conditions; Artan Sheshmani, Brooke Ullery, Ashvin Vishwanath for

special/topic/basic courses, spring 2017; Ling-Miao Chou for comments that improve the illustrations and moral support.The project is supported by NSF grants DMS-9803347 and DMS-0074329.

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Chien-Hao Liu dedicates this work to

Noel Brady, Hung-Wen Chang, Chongsun Chu, William Grosso, Pei-Ming Ho, Inkang Kim,

who enriched his years at U.C. Berkeley tremendously.∗

∗(From C.H.L.) It was an amazing time when I landed at Berkeley in the 1990s, following Thurston’s transition toMathematical Sciences Research Institute (M.S.R.I.). On the mathematical side, representing figures on geometry andtopology — 2- and 3-dimensional geometry and topology and related dynamical system (Andrew Casson, Curtis McMullen,William Thurston), 4-dimensional geometry and topology (Robion Kirby), 5-and-above dimensional topology (Morris Hirsch,Stephen Smale), algebraic geometry (Robin Hartshorne), differential and complex geometry (Wu-Yi Hsiang, ShoshichiKobayashi, Hong-Shi Wu), symplectic geometry and geometric quantization (Alan Weinstein), combinatorial and geometricgroup theory (relevant to 3-manifold study via the fundamental groups, John Stallings) — seem to converge at Berkeley.On the physics side, several first-or-second generation string-theorists (Orlando Alvarez, Korkut Bardakci, Martin Halpern)and one of the creators of supersymmetry (Bruno Zumino) were there. It was also a time when enumerative geometryand topology motivated by quantum field and string theory started to emerge and for that there were quantum invariantsof 3-manifolds (Nicolai Reshetikhin) and mirror symmetry (Alexandre Givental) at Berkeley. Through the topic coursesalmost all of them gave during these years on their related subject and the timed homeworks one at least attempted, onemay acquire a very broad foundation toward a cross field between mathematics and physics if one is ambitious and diligentenough.

However, despite such an amazing time and an intellectually enriching environment, it would be extremely difficult, ifnot impossible, to learn at least some good part of them without a friend — well, of course, unless one is a genius, which Iam painfully not. For that, on the mathematics side, I thank Hung-Wen (Weinstein and Givental’s student) who said helloto me in our first encounter at the elevator at the 9th floor in Evans Hall and influenced my understanding of many topics— particularly symplectic geometry, quantum mechanics and gauge theory — due to his unconventional background fromphysics, electrical engineering, to mathematics; Bill (Stallings’ student) and Inkang (Casson’s student) for suggesting me aweekly group meeting on Thurston’s lecture notes at Princeton on 3-manifolds

[Th] W.P. Thurston, The geometry and topology of three-manifolds, typed manuscript, Department of Mathematics,Princeton University, 1979.

which lasted for one and a half year and tremendously influenced the depth of my understanding of Thurston’s work; Noel(Stallings’ student) for a reading seminar on the very thought-provoking French book

[Gr] M. Gromov, Structures metrique pour les varietes riemanniennes, redige par J. Lafontaine et P. Pansu, TextMath. 1, Cedic/Fernand-Nathan, Paris, 1980.

for a summer. The numerous other after-class discussions for classes some of them and I happened to sit in shaped a largepart of my knowledge pool, even for today. Incidentally, thanks to Prof. Kirby, who served as the Chair for the GraduateStudents in the Department at that time. I remember his remark to his staff when I arrived at Berkeley and reported tohim after meeting Prof. Thurston: “We have to treat them [referring to all Thurston’s then students] as nice as our own”

On the physics side, thanks to Chongsun and Pei-Ming (both Zumino’s students) for helping me understand the verychallenging topic: quantum field theory, first when we all attended Prof. Bardakci’s course Phys 230A, Quantum FieldTheory, based closely on the book

[Ry] L.H. Ryder, Quantum field theory, Cambridge Univ. Press, 1985.

and a second time a year later when we repeated it through the same course given by Prof. Alvarez, followed by a readinggroup meeting on Quantum Field Theory guided by Prof. Alvarez — another person who I forever have to thank andanother event which changed the course of my life permanently. The former with Prof. Bardakci was a semester I hadto spend at least three-to-four days a week just on this course: attending lectures, understanding the notes, reading thecorresponding chapters or sections of the book, doing the homeworks, occasionally looking into literatures to figure outsome of the homeworks, and correcting the mistakes I had made on the returned homework. I even turned in the take-homefinal. Amusingly, due to the free style of the Department of Mathematics at Princeton University and the visiting studentstatus at U.C. Berkeley, that is the first of the only three courses and only two semesters throughout my graduate studentyears for which I ever honestly did like a student: do the homework, turn in to get graded, do the final, and in the end geta semester grade back. Special thanks to Prof. Bardakci and the TA for this course, Bogdan Morariu, for grading whateverI turned in, though I was not an officially registered student in that course.

That was a time when I could study something purely for the beauty, mystery, and/or joy of it. That was a time when

the future before me seemed unbounded. That was a time when I did not think too much about the less pleasant side of

doing research: competitions, publications, credits, · · · . That was a time I was surrounded by friends, though only limitedly

many, in all the best senses the word ‘friend’ can carry. Alas, that wonderful time, with such a luxurious leisure, is gone

forever!

Dynamics of D-branes, II: The Standard Action

0. Introduction and outline

In this sequel to D(11.1) (arXiv:1406.0929 [math.DG]), D(11.3.1) (arXiv:1508.02347 [math.DG]),D(13.1) (arXiv:1606.08529 [hep-th]) and D(13.2.1) (arXiv:1611.09439 [hep-th]) and along theline of our understanding of the basic structures on D-branes in Polchinski’s TASI 1996 Lec-ture Notes from the aspect of Grothendieck’s modern Algebraic Geometry initiated in D(1)

(arXiv:0709.1515 [math.AG]), we introduce a new action S(ρ,h;Φ,g,B,C)standard for D-branes that is

to D-branes as the (Brink-Di Vecchia-Howe/Deser-Zumino/)Polyakov action is to fundamentalstrings. This action depends both on the (dilaton field ρ, metric h) on the underlying topologyX of the D-brane world-volume and on the background (dilaton field Φ, metric g, B-field B,Ramond-Ramond field C) on the target space-time Y ; and is naturally a non-Abelian gaugedsigma model — based on maps ϕ : (XAz, E;∇) → Y from an Azumaya/matrix manifold XAz

with a fundamental module E with a connection ∇ to Y — enhanced by the dilaton term thatcouples (ϕ,∇) to (ρ,Φ), the B-coupled gauge-theory term that couples ∇ to B, and the Chern-

Simons/Wess-Zumino term that couples (ϕ,∇) to (B,C) in our standard action S(ρ,h;Φ,g,B,C)standard .

Before one can do so, one needs to resolve the built-in obstruction of pull-push of covarianttensors under a map from a noncommutative manifold to a commutative manifold. Such issuealready appeared in the construction of the non-Abelian Dirac-Born-Infeld action (D(13.1) ).In this note, we give a hierarchy of various admissible conditions on the pairs (ϕ,∇) that areenough to resolve the issue while being open-string compatible (Sec. 2). This improves ourunderstanding of admissible conditions beyond D(13.2.1). With the noncommutative analysisdeveloped in D(13.1), we develop further in this note some covariant differential calculus for suchmaps (Sec. 3) and use it to define the standard action for D-branes (Sec. 4). After promoting thesetting to a family version (Sec. 5), we work out the first variation — and hence the correspondingequations of motion for D-branes — of the standard action (Sec. 6) and the second variation ofthe kinetic term for maps and the dilaton term in this action (Sec. 7).

Compared with the non-Abelian Dirac-Born-Infeld action constructed in D(13.1) along thesame line, the current standard action is clearly much more manageable. Classically and math-ematically and in the special case where the background (Φ, B,C) on Y is set to vanish, thisnew theory is a merging of the theory of harmonic maps and a gauge theory (free to choose ei-ther a Yang-Mills theory or other kinds of applicable gauge theory) with a nilpotent type fuzzyextension. The current bosonic setting is the first step toward fermionic D-branes (cf. D(11.2):arXiv:1412.0771 [hep-th]) and their quantization as fundamental dynamical objects, in parallelto what happened for fundamental superstrings during 1976–1981; (the road-map at the end:‘Where we are’).

Convention. References for standard notations, terminology, operations and facts are(1) Azumaya/matrix algebra: [Ar], [Az], [A-N-T]; (2) sheaves and bundles: [H-L]; with con-nection: [Bl], [B-B], [D-K], [Ko]; (3) algebraic geometry: [Ha]; C∞ algebraic geometry: [Jo];(4) differential geometry: [Eis], [G-H-L], [Hi], [H-E], [K-N]; (5) noncommutative differentialgeometry: [GB-V-F]; (6) string theory and D-branes: [G-S-W], [Po2], [Po3].

· For clarity, the real line as a real 1-dimensional manifold is denoted by R1, while the fieldof real numbers is denoted by R. Similarly, the complex line as a complex 1-dimensionalmanifold is denoted by C1, while the field of complex numbers is denoted by C.

· The inclusion ‘R ⊂ C’ is referred to the field extension of R to C by adding√−1, unless

otherwise noted.

1

· All manifolds are paracompact, Hausdorff, and admitting a (locally finite) partition ofunity. We adopt the index convention for tensors from differential geometry. In particular,the tuple coordinate functions on an n-manifold is denoted by, for example, (y1, · · · yn).However, no up-low index summation convention is used.

· For this note, ‘differentiable’, ‘smooth’, and C∞ are taken as synonyms.

· matrix m vs. manifold of dimension m

· the Regge slope α′ vs. dummy labelling index α vs. covariant tensor α

· section s of a sheaf or vector bundle vs. dummy labelling index s

· algebra Aϕ vs. connection 1-form Aµ

· ring R vs. k-th remainder R[k] vs. Riemann curvature tensor Rijkl

· boundary ∂U of an open set U vs. partial differentiations ∂t, ∂/∂yi

· SpecR (:= prime ideals of R) of a commutative Noetherian ring R in algebraic geometryvs. SpecR of a Ck-ring R (:= Spec RR := Ck-ring homomorphisms R→ R)

· morphism between schemes in algebraic geometry vs. C∞-map between C∞-manifolds orC∞-schemes in differential topology and geometry or C∞-algebraic geometry

· group action vs. action functional for D-branes

· metric tensor g vs. element g′ in a group G vs. gauge coupling constant ggauge

· sheaves F , G vs. curvature tensor F∇, gauge-symmetry group Ggauge

· dilaton field ρ vs. representation ρgauge of a gauge-symmetry group Ggauge

· The ‘support’ Supp (F) of a quasi-coherent sheaf F on a scheme Y in algebraic geometryor on a Ck-scheme in Ck-algebraic geometry means the scheme-theoretical support of Funless otherwise noted; IZ denotes the ideal sheaf of a (resp. Ck-)subscheme of Z of a(resp. Ck-)scheme Y ; l(F) denotes the length of a coherent sheaf F of dimension 0.

· For a sheaf F on a topological space X, the notation ‘s ∈ F ’ means a local section s ∈ F(U)for some open set U ⊂ X.

· For an OX -module F , the fiber of F at x ∈ X is denoted F|x while the stalk of F at x ∈ Xis denoted Fx.

· coordinate-function index, e.g. (y1, · · · , yn) for a real manifold vs. the exponent of a power,e.g. a0y

r + a1yr−1 + · · · + ar−1y + ar ∈ R[y].

· The current Note D(13.3) continues the study in

[L-Y8] Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its firstvariation, and the equations of motion for D-branes — with remarks onthe non-Abelian Chern-Simons/Wess-Zumino term, arXiv:1606.08529 [hep-th].(D(13.1))

Notations and conventions follow ibidem when applicable.

2

Outline

0. Introduction.

1. Azumaya/matrix manifolds with a fundamental module and differentiable maps therefrom

· Azumaya/matrix manifolds with a fundamental module (XAz, E)

· When E is equipped with a connection ∇· Differentiable maps from (XAz, E)

· Compatibility between the map ϕ and the connection ∇

2. Pull-push of tensors and admissible conditions on (ϕ,∇)

2.1 Admissible conditions on (ϕ,∇) and the resolution of the pull-push issue

2.2 Admissible conditions from the aspect of open strings

3. The differential dϕ of ϕ and its decomposition, the three basic O CX -modules, induced structures,

and some covariant calculus

3.1 The differential dϕ of ϕ and its decomposition induced by ∇3.2 The three basic O C

X -modules relevant to Dϕ, with induced structures

3.2.1 The OAzX -valued cotangent sheaf T ∗X ⊗OX OAzX of X, and beyond

3.2.2 The pull-back tangent sheaf ϕ∗T∗Y3.2.3 The O C

X -module T ∗X ⊗OX ϕ∗T∗Y , where Dϕ lives

4. The standard action for D-branes

· The gauge-symmetry group C∞(Aut C(E))

· The standard action for D-branes

· The standard action as an enhanced non-Abelian gauged sigma model

5. Admissible family of admissible pairs (ϕT ,∇T )

· Basic setup and the notion of admissible families of admissible pairs (ϕT ,∇T )

· Three basic OXT -modules with induced structures

· Curvature tensors with ∂t and other order-switching formulae

· Two-parameter admissible families of admissible pairs

6. The first variation of the enhanced kinetic term for maps and ......

6.1 The first variation of the kinetic term for maps

6.2 The first variation of the dilaton term

6.3 The first variation of the gauge/Yang-Mills term and the Chern-Simons/Wess-Zumino term

6.3.1 The first variation of the gauge/Yang-Mills term

6.3.2 The first variation of the Chern-Simons/Wess-Zumino term for lower-dimensionalD-branes

6.3.2.1 D(−1)-brane world-point (m = 0)

6.3.2.2 D-particle world-line (m = 1)

6.3.2.3 D-string world-sheet (m = 2)

6.3.2.4 D-membrane world-volume (m = 3)

7. The second variation of the enhanced kinetic term for maps

6.1 The second variation of the kinetic term for maps

6.2 The second variation of the dilaton term

· Where we are

3

1 Azumaya/matrix manifolds with a fundamental module anddifferentiable maps therefrom

Basics of maps from an Azumaya/matrix manifold with a fundamental module needed for thecurrent note are collected in this section to fix terminology, notations, and conventions. Readersare referred to [L-Y1] (D(1)), [L-L-S-Y] (D(2)), [L-Y5] (D(11.1)) and [L-Y7] (D(11.3.1)) fordetails; in particular, why this is a most natural description of D-branes when Polchinski’s TASI1996 Lecture Note is read from the aspect of Grothendieck’s modern Algebraic Geometry. Seealso [H-W] and [Wi2].

Azumaya/matrix manifolds with a fundamental module (XAz, E)

From the viewpoint of Algebraic Geometry, a D-brane world-volume is a manifold equipped witha noncommutative structure sheaf of a special type dictated by (oriented) open strings.

Definition 1.1. [Azumaya/matrix manifold with fundamental module] Let X be a(real, smooth) manifold and E be a (smooth) complex vector bundle over X. Let

· OX be the structure sheaf of (smooth functions on) X,

· OCX := OX ⊗R C be its complexification,

· E be the sheaf of (smooth) sections of E, (it’s an OCX -module), and

· EndO CX

(E) be the endomorphism sheaf of E as an OCX -module

(i.e. the sheaf of sections of the endomorphism bundle End C(E) of E).

Then, the (noncommutative-)ringed topological space

XAz := (X,OAzX := EndO CX

(E))

is called an Azumaya manifold1(or synonymously, a matrix manifold to be more concrete tostring-theorists.) It is important to note that non-isomorphic complex vector bundles may giverise to isomorphic endomorphism bundles and from the string-theory origin of the setting, inwhich E plays the role of a Chan-Paton bundle on a D-brane world-volume, we always want torecord E as a part of the data in defining XAz. Thus, we call the pair (XAz, E) (or (XAz, E) inbundle notation) an Azumaya/matrix manifold with a fundamental module.

While it may be hard to visualize XAz geometrically, there in general is an abundant familyof commutative OX -subalgebras

OX ⊂ A ⊂ OAzXthat define an abundant family of C∞-schemes

XA := Spec R(A) ,

each finite and germwise algebraic over X. They may help visualize XAz geometrically.

Definition 1.2. [(commutative) surrogate of XAz] Such XA is called a (commutative)surrogate of (the noncommutative manifold) XAz. Cf.Figure 1-1.

4

(e)

(b)

( f )

(c)

X

XAz

X

(g)

(a)

(d)A

Figure 1-1. The noncommutative manifold XAz has an abundant collection of C∞-schemes as its commutative surrogates. See [L-Y8: Figure 2-1-1: caption] (D(13.1))for more details.

Without loss of generality, one may assume that X is connected. However even so, a surrogateXA of XAz in general is disconnected locally over X (and can be disconnected globally as well;cf. Figure 1-1). To keep track of this algebraically, recall the following definition:

Definition 1.3. [complete set of orthogonal idempotents] (Cf. e.g. [Ei].) Let R be an(associative, unital) ring, with the identity element 1. A set of elements e1, · · · , es ⊂ R iscalled a complete set of orthogonal idempotents if the following three conditions are satisfied

(1) idempotent e2i = ei, i = 1, · · · , s.

(2) orthogonal eiej = 0 for i 6= j.

(3) complete e1 + · · · + es = 1.

A complete set orthogonal idempotents e1, · · · , es is called maximal if no ei in the set can befurther decomposed into a summation ei = e′ + e′′ of two orthogonal idempotents.

Let OX ⊂ A ⊂ OAzX be a commutative OX -subalgebra of OAzX and XA the associate surrogateof XAz. Then, for U ⊂ X an open set, there is a unique maximal complete set of orthogonalidempotents e1, · · · , es of the C∞-ring A(U) and it corresponds to the set of connected

1Unfamiliar physicists may consult [Ar] for basics of Azumaya algebras; see also [Az] and [A-N-T]. Simplyput, an Azumaya manifold is topologically a smooth manifold but with a structure sheaf that has fibers Azumayaalgebras over C. These fibers are all isomorphic to a matrix ring Mr×r(C) (and hence the synonym matrixmanifold) for some fixed r but the isomorphisms involved are not canonical (and hence why the term ‘Azumayamanifold’ is more appropriate mathematically).

5

components of XA|U = Spec R(A(U)). Up to a relabelling, ei corresponds the function on XA(U)that is constant 1 on the i-th connected component and 0 on all other connected components.

Finally, we recall also the tangent sheaf and the cotangent sheaf of XAz.

Definition 1.4. [tangent sheaf, cotangent sheaf, inner derivations on XAz] The sheafof (left) derivations on OAzX is denoted by T∗XAz and is called the tangent sheaf of XAz. Thesheaf of Kahler differentials of OAzX is dented by T ∗XAz and is called the cotangent sheaf of XAz.T∗XAz is naturally a (left) OC

X -module while T ∗XAz is naturally a (left) OAzX -module. For ourpurpose, we treat both as OC

X -modules. There is a natural OCX -module homomorphism

OAzX −→ T∗XAzm 7−→ [m, · ] ,

where [m, · ] acts on OAzX by m′ 7→ [m,m′] := mm′ −m′m. The image of this homomorphismis called the sheaf/OC

X -module of inner derivations on OAzX and is denoted by Inn (OAzX ) orInn (XAz). The kernel of the above map is exactly the center OC

X · Id E , canonically identifiedwith OC

X , of OAzX . When the choice of a representative of an element of Inn (OAzX ) by an elementin OAzX is irrelevant to an issue, we’ll represent elements of Inn (OAzX ) simply by elements in OAzX .

When E is equipped with a connection ∇

From the stringy origin of the setting with E serving as the Chan-Paton bundle on the D-brane world-volume, E is equipped with a gauge field (i.e. a connection) created by masslessexcitations of open strings. Thus, let ∇ be a connection on E . Then ∇ induces a connectionD on OAzX := EndO C

X(E). With respect to a local trivialization of E , ∇ = d + A, where A

is an EndO CX

(E)-valued 1-form on X. Then D = d + [A , · ] on OAzX under the induced local

trivialization. As a consequence, D leaves the center OCX of OAzX invariant and restrict to the

usual differential d on OCX .

Once having the induced connection D on OAzX , one has then OCX -module homomorphism

T∗XC −→ T∗XAzξ 7−→ Dξ .

Lemma 1.5. [D-induced decomposition of T∗XAz] ([DV-M].) One has the short exact se-quence

0 −→ Inn (OAzX ) −→ T∗XAz −→ T∗XC −→ 0

split by the above map.

The following two lemmas address the issue of when an idempotent in OAzX can be constantunder a derivation ∈ T∗XAz.

Lemma 1.6. [(local) idempotent under D ] With the above notations, let U ⊂ X be anopen set, ξ a vector field on U , and e1, · · · , es be a complete set of orthogonal idempotents ofOAzX (U). Assume that, say, Dξe1 commutes with all ei, i = 1, · · · , s. Then Dξe1 = 0.

6

Proof. Since e21 = e1, (Dξe1)e1 + e1Dξe1 = Dξe1. If in addition Dξe1 and e1 commute, then one

has 2(Dξe1)e1 = Dξe1. The multiplication from the left by e1 gives then 2(Dξe1)e1 = (Dξe1)e1;i.e. (Dξe1)e1 = 0. If, furthermore, Dξe1 commutes also with all e2, · · · , es, then 0 = Dξ(eje1) =(Dξej)e1 + ejDξe1 = (Dξej)e1 + (Dξe1)ej , for j = 2, · · · , s. The multiplication from the left byej gives then (Dξe1)ej = 0, for j = 2, · · · , s. It follows that Dξe1 = (Dξe1)(e1 + · · · + es) = 0.

Lemma 1.7. [(local) idempotent under inner derivation] With the above notations, letU ⊂ X be an open set, m ∈ OAzX (U) represent an inner derivation of OAzX (U), and e1, · · · , esbe a complete set of orthogonal idempotents of OAzX (U). Assume that, say, [m, e1] commutes withall ei, i = 1, · · · , s. Then [m, e1] = 0.

Proof. Note that the proof of Lemma 1.6 uses only the Leibnitz rule property of Dξ on OAzX (U)and the commutativity property of Dξe1 with e1, · · · , es. Since [m, · ] satisfies also the Leibnizrule property on OAzX (U) and by assumption [m, e1] commutes with e1, · · · , es, the same proofgoes through.

The contraction EndO CX

(E) = E ⊗O CXE∨ → OC

X defines a trace map

Tr : OAzX −→ OCX .

One hasdTr = TrD ,

where d is the ordinary differential on OCX .

Differentiable maps from (XAz, E)

As a dynamical object in space-time, a D-brane moving in a space-time Y is realized by a mapfrom a D-brane world-volume to Y . Back to our language, we need thus a notion of a ‘map from(XAz, E ;∇) to Y ’ that is compatible with the behavior of D-branes in string theory.

Definition 1.8. [map from Azumaya/matrix manifold] Let X be a (real, smooth) mani-fold, E be a complex vector bundle of rank r over X, and (XAz, E) := (X,C∞(End C(E)), E) bethe associated Azumaya/matrix manifold with a fundamental module. A map (synonymously,differentiable map, smooth map)

ϕ : (XAz, E) −→ Y

from (XAz, E) to a (real, smooth) manifold Y is defined contravariantly by a ring-homomorphism

ϕ] : C∞(Y ) −→ C∞(End C(E)) .

Equivalently in terms of sheaf language, let OY be the structure sheaf of Y . Regard bothOY and OAzX as equivalence classes of gluing system of rings over the topological space Y andX respectively. Then the above ϕ] specifies an equivalence class of gluing systems of ring-homomorphisms over R ⊂ C

OY −→ OAzX ,

which we will still denote by ϕ].

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Through the Generalized Division Lemma a la Malgrange, one can show that ϕ] extends toa commutative diagram

OAzX OYϕ]oo

_

pr]Y

OX

pr]X

//?

OO

OX×Y ,

ϕ]hh

of equivalence classes of ring-homomorphisms (over R or R ⊂ C, whichever is applicable) betweenequivalence classes of gluing systems of rings, with

Aϕ OYf]ϕoo

_

pr]Y

OX

pr]X

//?

π]ϕ

OO

OX×Y ,

f]ϕ

hh

a commutative diagram of equivalence classes of ring-homomorphisms between equivalenceclasses of gluing systems of C∞-rings. Here, prX : X × Y → X and prY : X × Y → Y arethe projection maps, OX → OAzX follows from the inclusion of the center OC

X of OAzX , and

Aϕ := OX〈Imϕ]〉 = Im ϕ] .

(Cf. [L-Y7: Theorem 3.1.1] (D(11.3.1)).)In terms of spaces, one has the following equivalent diagram of maps

E

!!

XAz

ϕ

))

σϕ

Xϕfϕ

//w

fϕ ))πϕ

Y

X X × Y

prY

OOOO

prXoooo ,

where Xϕ is the C∞-schemeXϕ := Spec RAϕ

associated to Aϕ.

Definition 1.9. [graph of ϕ] The push-forward ϕ∗E =: Eϕ of E under ϕ is called the graph ofϕ. It is an OC

X×Y -module. Its C∞-scheme-theoretical support is denoted by Supp (Eϕ).

Definition 1.10. [surrogate of XAz specified by ϕ] The C∞-scheme Xϕ is called thesurrogate of XAz specified by ϕ.

Xϕ is finite and germwise algebraic over X and, by construction, it admits a canonicalembedding fϕ : Xϕ → X × Y into X × Y as a C∞-subscheme. The image is identical toSupp (Eϕ). Cf. Figure 1-2 and Figure 1-3.

8

Imϕ

fϕ

ϕ

ϕ

Y

X

Xϕ

XAz

Figure 1-2. A map ϕ : (XAz, E)→ Y specifies a surrogate Xϕ of XAz over X. Xϕ is aC∞-scheme that may not be reduced (i.e. it may have some nilpotent fuzzy structurethereon). It on one hand is dominated by XAz and on the other dominates and isfinte and germwise algebraic over X.

Compatibility between the map ϕ and the connection ∇

Up to this point, the map ϕ : (XAz, E)→ Y to Y and the connection∇ on E are quite independentobjects. A priori, there doesn’t seem to be any reason why they should constrain or influenceeach other at the current purely differential-topological level. However, when one moves onto address the issue of constructing an action functional for (ϕ,∇) as in [L-Y8] (D(13.1)), oneimmediately realizes that,

· Due to a built-in mathematical obstruction in the problem, one needs some compatibilitycondition between ϕ and ∇ before one can even begin the attempt to construct an actionfunctional for (ϕ,∇).

Furthermore, as a hindsight, that there needs to be a compatibility condition on (ϕ,∇) is alsoimplied by string theory:

· We need a condition on (ϕ,∇) to encode the stringy fact that the gauge field ∇ on theD-brane world-volume as ‘seen’ by open strings in Y through ϕ should be massless.

We address such compatibility condition on (ϕ,∇) systematically in the next section.

2 Pull-push of tensors and admissible conditions on (ϕ,∇)When one attempts to construct an action functional for a theory that involves maps from aworld-volume to a target space-time, one unavoidably has to come across the notion of ‘pullingback a (covariant) tensor, for example, the metric tensor or a differential form on the targetspace-time to the world-volume’. In the case where only maps from a commutative world-volume to a (commutative) space-time are involved, this is a well-established standard notionfrom differential topology. However, in a case, like ours, where maps from a noncommutativeworld-volume to a (commutative) space-time is involved,

ϕ : Space (S) −→ Space(R) ,

9

X

πϕ

Xϕ

nd X ( )

Azumaya cloudYϕ

fϕ

X

Y

ϕΓSupp ( ) =

=

X Y

pr2

pr1

map from Azumaya manifold

Fourier-Mukai transform

Xϕ

Figure 1-3. The equivalence between a map ϕ from an Azumaya manifold with afundamental module (X,OAzX := EndO C

X(E), E) to a manifold Y and a special kind of

Fourier-Mukai transform E ∈ Mod C(X × Y ) from X to Y . Here, Mod C(X × Y ) isthe category of O C

X×Y -modules.

10

with the accompanying contravariant ring-homomorphism

S ←− R : ϕ] ,

there is a built-in mathematical obstruction to such a notion. Here, S is an (associative, unital)noncommutative ring, R is a (associative, unital) commutative ring, and Space (S) and Space (R)are the topological spaces whose function rings are S and R respectively.

For the noncommutative ring S, its (standard and functorial-in-the-category-rings) bi-S-module of Kahler differentials is naturally defined to be

ΩKahlerS := Span (S,S)ds | s ∈ S)/(d(ss′)− (ds)s′ − sds′ | s, s′ ∈ S)

while for the commutative ring R its (standard and functorial-in-the-category-of-commutative-rings) (left) R-module of Kahler differentials is naturally defined to be

ΩKahlerR := SpanRdr | r ∈ R/(d(rr′)− r′dr − rdr′ | r, r′ ∈ R) ,

with the convention that rdr′ = (dr′)r to turn it to a bi-R-module as well. Treating R as a ring(that happens to be commutative), it has also the (standard and functorial-in-the-category-rings)bi-R-module of Kahler differentials

Ωnc,KahlerR := Span (R,R)dr | r ∈ R)/(d(rr′)− (dr)r′ − rdr′ | r, r′ ∈ R)

exactly like ΩKahlerS for S. There is a built-in tautological quotient homomorphism as bi-R-

modulesΩnc,KahlerR −→−→ ΩKahler

R

r1(dr)r2 7−→ r1r2 dr ,

whose kernel is generated by rdr′−(dr′)r | r, r′ ∈ R. Given the map ϕ : Space (S)→ Space (R),one has the following built-in diagram

ΩKahlerS Ωnc,Kahler

R

ϕ∗oo

ΩKahlerR ,

where ϕ∗(r1(dr)r2) = ϕ](r1)d(ϕ](r))ϕ](r2). The issue is now whether one can extend the abovediagram to the following commutative diagram

ΩKahlerS Ωnc,Kahler

R

ϕ∗oo

ΩKahlerR

ϕ∗?

hh

.

The answer is No, in general. See, e.g., [L-Y5: Example 4.1.20] (D(11.1)) for an explicit coun-terexample. When R is a C∞-ring, e.g. the function-ring C∞(Y ) of a smooth manifold Y , thenthe R-module ΩR of differentials of R is a further quotient of the above module ΩKahler

R of Kahlerdifferentials by additional relations generated by applications of the chain rule on the transcen-dental smooth operations in the C∞-ring structure of R ([Jo]; cf. [L-Y5: Sec. 4.1] (D(11.1))).The issue becomes even more involved. In particular, as the counterexample ibidem shows

· [built-in mathematical obstruction of pullback] For a map ϕ : (XAz, E) → Y ,there is no way to define functorially a pull-back map T ∗Y → T ∗XAz that takes a(covariant) 1-tensor on Y to a 1-tensor on XAz. As a consequence, there is no functorialway to pull back a (covariant) tensor on Y to a tensor on XAz.

11

Before the attempt to construct an action functional that involves such maps ϕ, one has toresolve the above obstruction first. In [L-Y8] (D(13.1)), we learned how to use the connection ∇to impose a natural admissible condition on ϕ so that the above obstruction is bypassed throughthe surrogate Xϕ of XAz specified by ϕ. With the lesson learned therefrom and further thoughtbeyond [L-Y9] (D(13.2.1)), we propose (Sec. 2.1) in this section a still-natural-but-much-weakeradmissible condition on (ϕ,∇) that bypasses even the surrogate Xϕ but is still robust enoughto construct naturally a pull-push map we need on tensors. It turns out that this much weakeradmissible condition remains to be compatible with open strings (Sec. 2.2).

2.1 Admissible conditions on (ϕ,∇) and the resolution of the pull-push issue

A hierarchy of admissible conditions on (ϕ,∇) is introduced. A theorem on how even the weakestadmissible condition in the hierarchy can resolve the above obstruction in our case is proved.

Three hierarchical admissible conditions

Definition 2.1.1. [admissible connection on E] Let ϕ : (XAz, E) → Y be a map. For aconnection ∇ on E , let D be its induced connection on OAzX . A connection ∇ on E is called

(∗1)-admissible to ϕ if DξAϕ ⊂ Comm (Aϕ);

(∗2)-admissible to ϕ if DξComm (Aϕ) ⊂ Comm (Aϕ);

(∗3)-admissible to ϕ if DξAϕ ⊂ Aϕ

for all ξ ∈ T∗X. Here, Comm (Aϕ) denotes the commutant of Aϕ in OAzX .When ∇ is (∗1)-admissible to ϕ, we will take the following as synonyms:

· (ϕ,∇) is an (∗1)-admissible pair,

· ϕ is (∗1)-admissible to ∇,

· ϕ : (XAz, E ;∇)→ Y is (∗1)-admissible.

Similarly, for (∗2)-admissible pair (ϕ,∇) and (∗3)-admissible pair (ϕ,∇), ... , etc..

Lemma 2.1.2. [hierarchy of admissible conditions]

Admissible Condition (∗3) =⇒ Admissible Condition (∗2) =⇒ Admissible Condition (∗1) .

Proof. Admissible Condition (∗3) says that the OX -subalgebra Aϕ ⊂ OAzX is invariant underD-parallel transports along paths on X. Since D-parallel transports on OAzX are algebra-isomorphisms, if Aϕ is D-invariant, the OX -subalgebra Comm (Aϕ) of OAzX must also be D-invariant since it is determined by Aϕ fiberwise algebraically. In other words, Admissible Con-dition (∗3) =⇒ Admissible Condition (∗2).

Since Aϕ is commutative, Aϕ ⊂ Comm (Aϕ.). Thus, the inclusion D·Aϕ ⊂ D·Comm (Aϕ)always holds. This implies that Admissible Condition (∗2) =⇒ Admissible Condition (∗1).

Definition 2.1.3. [strict admissible connection on E] Continuing Definition 2.1.1. Let F∇be the curvature tensor of ∇. It is an OAzX -valued 2-form on X. Then, for ·= 1, 2, 3, ∇ is calledstrictly (∗·)-admissible to ϕ if

12

· ∇ is (∗·)-admissible to ϕ and F∇ takes values in Comm (Aϕ) ⊂ OAzX .

In this case, (ϕ,∇) is said to be a strictly (∗·)-admissible pair.

Clearly, the same hierarchy holds for strict admissible conditions:

strict (∗3) =⇒ strict (∗2) =⇒ strict (∗1) .

The Strict (∗3)-Admissible Condition on (ϕ,∇) was introduced in [L-Y8: Definition 2.2.1](D(13.1)) to define the Dirac-Born-Infeld action for (ϕ,∇).

Lemma 2.1.4. [commutativity under admissible condition] Let ϕ : (XAz, E ;∇) → Ybe a map. (1) If (ϕ,∇) is (∗1)-admissible, then [Dξϕ

](f1), ϕ](f2)] = 0 for all f1, f2 ∈ C∞(Y )and ξ ∈ T∗X. (2) If (ϕ,∇) is (∗2)-admissible, then [Dξ1ϕ

](f1), Dξ2ϕ](f2)] = 0 for all f1, f2 ∈

C∞(Y ) and ξ1, ξ2 ∈ T∗X.

Proof. Statement (1) is the (∗1)-Admissible Condition itself.For Statement (2), let f1, f2 ∈ C∞(Y ) and ξ1, ξ2 ∈ T∗X. Then [Dξϕ

](f1), ϕ](f2)] = 0 sinceAϕ ⊂ Comm (Aϕ). Thus, applying Dξ2 to both sides,

[Dξ2Dξ1ϕ](f1), ϕ](f2)] + [Dξ1ϕ

](f1), Dξ2ϕ](f2)] = 0 .

The (∗2)-Admissible Condition implies that

Dξ2Dξ1ϕ](f1) ∈ Comm (Aϕ) .

And, hence, [Dξ2Dξ1ϕ](f1), ϕ](f2)] = 0. Statement (2) follows.

Resolution of the pull-push issue under Admissible Condition (∗1)

The current theme is devoted to the proof of the following theorem:

Theorem 2.1.5. [pull-push under (∗1)-admissible (ϕ,∇)] Let (ϕ,∇) be (∗1)-admissible.Then the assignment

ϕ : ΩC∞(Y ) −→ ΩC∞(X) ⊗C∞(X) C∞(End C(E))

f1df2 7−→ ϕ](f1)Dϕ](f2)

is well-defined.

The study in [L-Y8: Sec. 4] (D(13.1)) allows one to express ϕ(df) locally explicit enoughso that one can check that ϕ is well-defined when (ϕ,∇) is (∗1)-admissible. Note that, withLemma 2.1.2, this implies that if (ϕ,∇) is either (∗2)- or (∗3)-admissible, then ϕ is also well-defined. We now proceed to prove the theorem.

13

Lemma 2.1.6. [local expression of ϕ(df) for (∗1)-admissible (ϕ,∇), I] Let (ϕ,∇) be(∗1)-admissible; i.e. DξAϕ ⊂ Comm (Aϕ) for all ξ ∈ T∗X. Let U ⊂ X be a small enough openset so that ϕ(UAz) is contained in a coordinate chart of Y , with coordinate y = (y1, · · · , yn).For f ∈ C∞(Y ), recall the germwise-over-U polynomial Rf [1] in (y1, · · · , yn) with coefficientsin OAzU from [L-Y8: Sec. 4 & Remark/Notation 4.2.3.5] (D.13.1). Then, for ξ a vector field onU and f ∈ C∞(Y ), and at the level of germs over U ,

(ϕ(df))(ξ) = Rf [1]∣∣yd Dξ(ϕ](yd)), for all multi-degree d in Rf [1]

.

Here, for a multiple degree d = (d1, , · · · , dn), di ∈ Z≥0, yd := (y1)d1 · · · (yn)dn andyd Dξ(ϕ

](yd)) means ‘replacing yd by Dξ(ϕ](yd))’.

Proof. Denote the coordinate chart of Y in the Statement by V . Let prX : X × Y → X,prY : X × Y → Y be the projection maps. Recall the induced ring-homomorphismϕ] : C∞(X ×Y )→ C∞(End C(E)) over R ⊂ C and the graph Eϕ of ϕ and its support Supp (Eϕ)

on X × Y . Denote pr]Y (f) ∈ C∞(X × Y ) still by f when there is no confusion. For clarity, weproceed the proof of the Statement in three steps.

Step (1) How Rf [1] is constructed in [L-Y8: Sec. 4] (D(13.1)) For any p ∈ U , let p ∈ U ′ ⊂ Ube a neighborhood of p in U over which the Generalized Division Lemma a la Malgrange is appliedto f on a neighborhood U ′×V ′ of (p×V ∩ Supp (Eϕ))red =: q1, · · · , qs in (X ×Y )/X with

respect to the characteristic polynomials χ(i)ϕ := det(yi · Id r×r−ϕ](yi)) ∈ C∞(U ′)[y1, · · · , yn] ∈

C∞(U ′ × V ), i = 1, · · · , n. Passing to a smaller open subset if necessary, one may assume thatV ′ is a disjoint union V ′1 ∪ · · · ∪V ′s with U ′×V ′k a neighborhood of qk and the closure V1, · · · , Vsare all disjoint from each other. Let 1(k) ∈ C∞(Y ) be a smooth functions on Y that takes thevalue 1 on V ′k and the value zero on V ′k′ , k

′ 6= k, k = 1, · · · , s. (Cf. [L-Y8: Sec. 4.2.3] (D(13.1)).)Then

f |U ′×V ′ =s∑

k=1

1(k) ·(∑

d

cf ;kd yd +

n∑i,j=1

Qf ;k(i,j)χ

(i)ϕ χ

(j)ϕ

)for some

· cf ;kd ∈ C∞(U ′) for all k and d, and

∑d c

f ;kd yd ∈ C∞(U ′)[y1, · · · , yn] for all k,

· Qf ;k(i,j) ∈ C

∞(U ′ × V ′k) for all k and i, j.

In terms of this, over U ′,

Rf [1] =s∑

k=1

ϕ](1(k))∑d

cf ;kd yd

andϕ](f)

∣∣U ′

= Rf [1]∣∣yd ϕ](yd) for all multi-degree d in Rf [1]

since ϕ](χ(i)ϕ ) = 0 for i = 1, · · · , n, ϕ](f) = ϕ](pr]Y (f)).

Step (2) ϕ](1(k))sk=1 as the maximal complete set of orthogonal D-parallel idempotents/U ′

Since ϕ](f) depends only on the restriction of f , regarded on X × Y , to Supp (Eϕ), one has

ϕ](1(1) + · · · + 1(s)) = ϕ](1X×Y ) = Id E|U′

over U ′ ⊂ X. Since, in addition, 1 2(k) = 1(k) for all k, 1(k) 1(k′) = 0 for all k 6= k′, and

s = the number of the connected components of Xϕ|U ′ for U ′ small enough, the collectionϕ](1(1)), · · · , ϕ](1(s)) gives the maximal complete set of orthogonal idempotents in Aϕ|U ′ .

14

Furthermore, since DξAϕ ⊂ Comm (Aϕ) for all ξ ∈ T∗X, Dξϕ](1(k)) and ϕ](1(k′)) commute

for all k, k′ = 1, · · · , s. It follows from Lemma 1.6 that

Dξϕ](1(1)) = · · · = Dξϕ

](1(s)) = 0

for all ξ. In other words, ϕ](1(1)), · · · , ϕ](1(s)) are D-parallel over U ′.

Step (3) The evaluation of (ϕ(df))(ξ) over U ′ We are now ready to evaluate the OAzX -valued derivation Dξϕ on f , locally and germwise over U ′. Step (1) and Step (2) together implythat, for ξ ∈ T∗X and over U ′,

(ϕ(df))(ξ) := Dξ(ϕ](f))

=s∑

k=1

ϕ](1(k))∑d

(ξcf ;kd )ϕ](yd) +

s∑k=1

ϕ](1(k))∑d

cf ;kd Dξ(ϕ

](yd))

= Term (I) + Term (II)

since Dξϕ](1(k)) = 0 for k = 1, · · · , s.

Note thatTerm (II) = Rf [1]

∣∣yd Dξ(ϕ](yd)), for all multi-degree d in Rf [1]

.

It remains to prove that Term (I) vanishes. But this is the situation studied in [L-Y8: Proposition4.2.3.1:Proof] (D(13.1)). In essence, since

f =s∑

k=1

1(k) ·(∑

d

cf ;kd yd +

n∑i,j=1

Qf ;k(i,j)χ

(i)ϕ χ

(j)ϕ

)on U ′ × V ′ and f on (X × Y )/X is independent of X, one has

Term (I) = ϕ](∑

k

1(k)

∑d

(ξcf ;k

d

)yd)

= ϕ](ξf) = 0 .

Here we denote the canonical lifting of ξ ∈ T∗X to T∗(X × Y ), via the product structure ofX × Y , by the same notation ξ.

This completes the proof.

Lemma 2.1.7. [local expression of ϕ(df) for (∗1)-admissible (ϕ,∇), II] Let (ϕ,∇) be(∗1)-admissible. Continuing the setting and notations in Lemma 2.1.6.

Then, locally,

(ϕ(df))(ξ) =

n∑i=1

(ϕdyi)(ξ)⊗ ∂∂yif =

n∑i=1

(Dξϕ

](yi) · ϕ]( ∂f∂yi)).

Here · is the multiplication in the ring C∞(End C(E)) (and will be omitted later when there isno sacrifice to clarity).

15

Proof. This is a consequence of Lemma 2.1.6. Continuing the setup in the Statement and the

proof thereof. Then, since ϕ](χ(i)ϕ ) = 0 for i = 1, · · · , n, one has

ϕ]( ∂∂yi f

)= ϕ]

(∂∂yiRf [1]

)= Rf [1]

∣∣yd ϕ]

(∂

∂yiyd), for all multi-degree d in Rf [1]

.

Since (ϕ,∇) is (∗1)-admissible, Dξϕ](yi) and ϕ](yj) commute for i, j = 1, · · · , n. It follows that

n∑i=1

Dξϕ](yi) · ϕ]( ∂

∂yif) = Rf [1]|yd Dξ(ϕ](yd)), for all multi-degree d in Rf [1] .

Which is ϕ(df) by Lemma 2.1.6. This proves the lemma.

Proof of Theorem 2.1.5. We now check in two steps that ϕ is well-defined. Note that we onlyneed to do so locally over X. Thus, let U ⊂ X be the open set in Lemma 2.1.6 such that ϕ(UAz)is contained in a coordinate chart V of Y , with the coordinate (y1, · · · , yn). Lemma 2.1.7 impliesthen that the following assignment is the restriction of ϕ to over U and hence is independentof the local coordinate (y1, · · · , yn) on V :

ϕ : ΩC∞(V ) −→ ΩC∞(U) ⊗C∞(U) C∞(End C(E|U ))

f1df2 7−→ ϕ](f1) ·∑n

i=1

(ϕ](∂f2

∂yi)·Dϕ](yi)

).

Here, we use again the fact that (ϕ,∇) is (∗1)-admissible so that the summand Dϕ](yi) ·ϕ]( ∂f∂yi

)in Lemma 2.1.7 is equal to ϕ]

(∂f2

∂yi

)· Dϕ](yi) here, with f replaced by f2. It remains to show

that ϕ is compatible with (a) the commutative Leibniz rule and (b) the chain-rule identitiesfrom the C∞-ring structure of C∞(V ).

(a) The commutative Leibniz rule For f1, f2 ∈ C∞ ∈ C∞(V ), one has

d(f1f2) − f2 df1 − f1 df2 = 0

in ΩC∞(V ). Under ϕ, one has

ϕ(d(f1f2) − f2 df1 − f1 df2

)= Dϕ](f1f2) − ϕ](f2)Dϕ](f1) − ϕ](f1)Dϕ](f2) = 0

sinceD(ϕ](f1f2)

)= D

(ϕ](f1)ϕ](f2)

)= (Dϕ](f1))ϕ](f2) + ϕ](f1)Dϕ](f2) ,

which isϕ](f2)Dϕ](f1) + ϕ](f1)Dϕ](f2)

for (ϕ,∇) (∗1)-admissible.

(b) The chain-rule identities from the C∞-ring structure Let ζ ∈ C∞(Rl), l ∈ Z≥1 andf1, · · · , fl ∈ C∞(V ). Then, one has

d(ζ(f1, · · · , fl)

)−

l∑k=1

(∂kζ)(f1, · · · , fl) dfk = 0

16

in ΩC∞(V ). Here, ∂kζ is the partial derivative of ζ ∈ C∞(Rl) with respect to its k-th argument.Under ϕ, one has

ϕ(d(ζ(f1, · · · , fl)

)−

l∑k=1

(∂kζ)(f1, · · · , fl) dfk)

= Dϕ](ζ(f1, · · · , fl)

)−

l∑k=1

ϕ]((∂kζ)(f1, · · · , fl)

)Dϕ](fk) = 0

since, by Lemma 2.1.7 and the (∗1)-admissibility of (ϕ,∇),

Dϕ](ζ(f1, · · · , fl)

)=

n∑i=1

Dϕ](yi)⊗ ∂∂yi ζ(f1, · · · , fl)

=n∑i=1

Dϕ](yi)⊗l∑

k=1

(∂kζ)(f1, · · · , fl)∂fk∂yi

=l∑

k=1

Dϕ](fk)⊗ (∂kζ)(f1, · · · , fl)

=l∑

k=1

Dϕ](fk)ϕ]((∂kζ)(f1, · · · , fl)

)=

l∑k=1

ϕ]((∂kζ)(f1, · · · , fl)

)Dϕ](fk) .

This completes the proof of Theorem 2.1.5

The pull-push ϕ on tensor product ΩC∞(Y ) ⊗C∞(Y ) · · · ⊗C∞(Y ) ΩC∞(Y )

Having the well-defined

ϕ : ΩC∞(Y ) −→ ΩC∞(X) ⊗C∞(X) C∞(End C(E)) ,

it is natural to consider the extension of ϕ to a correspondence between tensor products

ϕ : ⊗kC∞(Y )ΩC∞(Y ) −→(⊗kC∞(X) ΩC∞(X)

)⊗C∞(X) C

∞(End C(E))

f0 df1 ⊗ · · · ⊗ dfk 7−→ ϕ](f0)Dϕ](f1)⊗ · · · ⊗Dϕ](fk) .

Here, the tensor Dϕ](f1)⊗ · · · ⊗Dϕ](fk) is defined to the tensors of the underlying 1-forms inΩC∞(X) and multiplication of the coefficients in C∞(End C(E)) from each factor. Explicitly, interms of a local coordinate (x1, · · · , xm) on a chart U ⊂ X,

ϕ](f0)Dϕ](f1)⊗ · · · ⊗Dϕ](fk)

=m∑

µ1, ··· , µk=1

ϕ](f0)D∂/∂xµ1ϕ](f1) · · · D∂/∂xµkϕ

](fk) dxµ1 ⊗ · · · ⊗ dxµk .

Lemma 2.1.8. [pull-push of (covariant) tensor] For (ϕ,∇) (∗1)-admissible, the aboveextension of ϕ to covariant tensors is well-defined.

17

Proof. In ⊗kC∞(Y )ΩC∞(Y ), one has the identities

f0 df1 ⊗ df2 ⊗ · · · ⊗ dfk = df1f0 ⊗ df2 ⊗ · · · ⊗ dfk= df1 ⊗ f0df2 ⊗ · · · ⊗ dfk = · · · · · ·= df1 ⊗ · · · ⊗ dfk−1f0 ⊗ dfk = df1 ⊗ · · · ⊗ f0dfk = df1 ⊗ · · · ⊗ dfkf0 .

Since the (∗1)-Admissible Condition implies that ϕ](f0) commutes with all of Dϕ](f1), · · · ,Dϕ](fk), the parallel identities

ϕ](f0)Dϕ](f1)⊗Dϕ](f2)⊗ · · · ⊗Dϕ](fk) = Dϕ](f1)ϕ](f0)⊗Dϕ](f2)⊗ · · · ⊗Dϕ](fk)

= Dϕ](f1)⊗ ϕ](f0)Dϕ](f2)⊗ · · · ⊗Dϕ](fk) = · · · · · ·

= Dϕ](f1)⊗ · · · ⊗Dϕ](fk−1)ϕ](f0)⊗Dϕ](fk)

= Dϕ](f1)⊗ · · · ⊗ ϕ](f0)Dϕ](fk) = Dϕ](f1)⊗ · · · ⊗Dϕ](fk)ϕ](f0)

hold in(⊗kC∞(X) ΩC∞(X)

)⊗C∞(X) C

∞(End C(E)). This proves the lemma.

Note that for a (∗1)-admissible map ϕ : (XAz, E ;∇) → Y , since DξAϕ ⊂ Comm(Aϕ) forall ξ ∈ T∗X and Comm (Aϕ) is itself a (possibly noncommutative) OC

X -subalgebra of OAzX , thepull-push ϕα of a (covariant) tensor α on Y to X is indeed Comm (Aϕ)-valued.

Example 2.1.9. [pull-push of 2-tensor under (∗1)-admissible (ϕ,∇)] Let ϕ : (XAz, E ;∇)→Y be a (∗1)-admissible map and α =

∑i,j αijdy

i⊗ yj be a 2-tensor on Y . Then, with respect to

local coordinates (x1, · · · , xn) on X and (y1, · · · , yn) on Y ,

ϕα =m∑

µ,ν=1

( n∑i,j=1

ϕ](αij)D ∂∂xµ

ϕ](yi)D ∂∂xν

ϕ](yj))dxµ ⊗ dxν .

Since in general D∂/∂xµϕ](yi)D∂/∂xνϕ

](yj) 6= D∂/∂xνϕ](yj)D∂/∂xµϕ

](yi), ϕ does not take asymmetric 2-tensor on Y to a Comm (Aϕ)-valued symmetric 2-tensor on X, nor an antisym-metric 2-tensor on Y to a Comm (Aϕ)-valued antisymmetric 2-tensor on X. However, afterthe post-composition with the trace map Tr : OAzX → OC

X , Trϕ does take a symmetric (resp.antisymmetric) 2-tensor on X to an OC

X -valued symmetric (resp. antisymmetric) 2-tensor on X.

Example 2.1.10. [pull-push of higher-rank tensor under (∗1)-admissible (ϕ,∇)] Con-tinuing Example 2.1.9. For α a (covariant) tensor on Y of rank ≥ 3, the trace map no longerhelp bring symmetric (resp. antisymmetric) tensors to symmetric (resp. antisymmetric) tensors.

The situation gets better for a map ϕ : (XAz, E ;∇) → Y that satisfies the stronger (∗2)-Admissible Condition: DξComm (Aϕ) ⊂ Comm (Aϕ) for ξ ∈ T∗X.

Lemma 2.1.11. [pull-push of tensor under (∗2)-admissible (ϕ,∇)] Let ϕ : (XAz, E ;∇)→Y be a (∗2)-admissible map. Then ϕ takes a symmetric (resp. antisymmetric) tensor on Y toa Comm (Aϕ)-valued symmetric (resp. antisymmetric) tensor on X.

18

Proof. In terms of local coordinates (x1, · · · , xn) on X and (y1, · · · , yn) on Y , a (covariant)k-tensor

α =∑i1, ··· ik

αi1 ··· ikdyi1 ⊗ · · · ⊗ dyik

on Y is pull-pushed to a Comm (Aϕ)-valued k-tensor

ϕα =

m∑µ1, ··· , µk=1

( n∑i1, ··· , ik=1

ϕ](αi1 ··· ik)D ∂∂xµ1

ϕ](yi1) · · · D ∂∂xµk

ϕ](yik))dxµ1 ⊗ · · · ⊗ dxµk .

on X. It follows from Lemma 2.1.4 that, under the (∗2)-Admissible Condition, all the factors

ϕ](αi1 ··· ik) , D ∂∂xµ1

ϕ](yi1) , · · · , D ∂∂xµk

ϕ](yik)

in a summand commute among themselves. This implies, in particular, that ϕ now takes asymmetric (resp. antisymmetric) tensor on Y to a Comm (Aϕ)-valued symmetric (resp. antisym-metric) tensor on X.

Let∧• T ∗Y be the sheaf of differential forms on Y . The same proof of Lemma 2.1.11 gives

also

Lemma 2.1.12. [ϕ and ∧] Let ϕ : (XAz, E ;∇) → Y be a (∗2)-admissible map. For α, β ∈∧• T ∗Y , define the wedge productϕα ∧ ϕβ

of ϕα, ϕβ ∈ (∧• T ∗X)C ⊗O C

XOAzX by applying the wedge product to the differential forms on

X and multiplication to the OAzX -valued coefficients. Then,

ϕ(α ∧ β) = (ϕα) ∧ (ϕβ) .

Remark 2.1.13. [admissible condition and Ramond-Ramond field ] While the current note willtake (ϕ,∇) to be (∗1)-admissible most of the time, Example 2.1.9, Example 2.1.10, Lemma 2.1.11and Lemma 2.1.12 together suggest that when the coupling of D-brane to Ramond-Ramondfields is taken into account, the more natural admissible condition on (ϕ,∇) is the stronger(∗2)-Admissible Condition.

2.2 Admissible conditions from the aspect of open strings

We address in this subsection the implication of Admissible Condition (∗1) on (ϕ,∇) to themass of the connection ∇ from the aspect of open strings in the target-space Y .

Let ϕ : (XAz, E ;∇) → Y be a (∗1)-admissible map. Recall the surrogate Xϕ := Spec RAϕof XAz specified by ϕ and the built-in dominant morphism πϕ : Xϕ → X; cf.Figure 1-2. Forx ∈ X, let e1, · · · , es ⊂ Aϕ, x the maximal complete set of orthogonal idempotents in thestalk of Aϕ at x. Then, by (∗1)-admissibility of (ϕ,∇) and Lemma 1.6,

Dξe1 = · · · · · · = Dξes = 0

for all ξ ∈ (T∗X)x. It follows that

19

Lemma 2.2.1. [covariantly invariant decomposition of stalks of E] For any x ∈ X, thedecomposition

Ex = e1Ex + · · · + esExis invariant under ∇. I.e. ∇ξ(ekEx) ⊂ ekEx, for k = 1, · · · , s and all ξ ∈ (T∗X)x.

As a consequence, the connection ∇ on Ex induces a connection ∇(k) on each direct summandekEx of Ex and one has the direct-sum decomposition

(Ex,∇) = (e1Ex,∇(1))⊕ · · · ⊕ (esEx,∇(s)) .

On the other hand, the maximal complete set of orthogonal idempotents e1, · · · , es ⊂ Aϕ, xcorresponds canonically and bijectively to the set of connected components of the germ Xϕ, x ofXϕ over x ∈ X:

Xϕ, x = X(1)ϕ, x t · · · tX(s)

ϕ, x .

Through the built-in inclusion Aϕ, x ⊂ OAzX x, Ex as the fundamental OAzX, x-module is canonicallyan Aϕ, x-module as well. Since ekel = 0 for k 6= l, as an Aϕ, x-module the direct summand ekExis supported exactly on X

(k)ϕ, x, for k = 1, · · · , s. The above decomposition of (E ,∇) says then

geometrically and in terms of physics terminology that the gauge field ∇ on E has no components

that mixes ekEx on X(k)ϕ, x and elE on X

(l)ϕ, x for some k 6= l; cf. Figure 2-2-1.

Recall now the string-theory origin of D-branes:

· A D-brane is where the end-points of an open string stick to.

· Excitations of open strings create fields on the D-brane.

· As the tension of open strings are constant, the mass of an open string — and hene fieldsit creates on the D-brane— is proportional to its length. Open strings with arbitrarilysmall length create massless fields on the brane while open strings with length boundedaway from zero create massive fields on the brane.

That the germ (Xϕ, x, Ex;∇) over any x ∈ X is decomposable in accordance with the connected-component decomposition of Xϕ, x says that ∇ must be created by open-strings of arbitrarilysmall length, rather than by those of length bounded away from zero. In other words, ∇ ismassless.

In summary:

Corollary 2.2.2. [(∗1)-Admissible Condition implies massless of ∇] For a (∗1)-admissiblemap ϕ : (XAz, E ;∇) → Y , the gauge field ∇ on the Chan-Paton sheaf E on the D-brane (orD-brane world-volume) XAz is massless from the aspect of open strings in the taget-space (ortarget-space-time) Y .

By Lemma 2.1.2, the same holds for (∗2)-admissible maps and (∗3)-admissible maps as well.

3 The differential dϕ of ϕ and its decomposition, the three basicOCX-modules, induced structures, and some covariant calculus

At the classical level Polyakov string or its generalization, a sigma model, is a theory of harmonicmaps on the mathematical side. In this section we construct all the building blocks to generalizethe existing theory of harmonic maps to a theory of maps ϕ : (XAz, E ;∇) → Y , which describeD-branes. It will turn out that both the connection ∇ and the Admissible Condition (∗1) chosenare needed to build up a mathematically sound theory for such maps ϕ.

20

X

XAz

φ ∇( , )general

φ

φ ∇( , ) that satisfiesAdmissible Condition (∗ )

D-brane world-volumewith a gauge field

Image brane in space-time Ywith the push-forward gauge field via φ

Y

φIm

φIm

φImY

1

Figure 2-2-1. When (ϕ,∇) is (∗1)-admissible, the gauge field ∇ on the Chan-Patonsheaf E on any small neighborhood U of x ∈ X localizes at each connected branchof ϕ(UAz) from the viewpoint of open strings in Y . In other words, ∇ is masslessfrom the open-string aspect. In the illustration, the noncommutative space XAz isexpressed as a noncommutative cloud shadowing over its underlying topology X, theconnection ∇ on E over X is indicated by a gauge field on X. Both the gauge fieldon X and how open strings “see” it in Y are indicated by squiggling arrows . Thesituation for a general (ϕ,∇) (cf. top) and a (ϕ,∇) satisfying Admissible Condition(∗1) (cf. bottom) are compared. From the open-string aspect, in the former situation∇ can have both massless components (which are local fields from the open-stringand target-space viewpoint) and massive components (which become nonlocal fieldsfrom the open-string and target-space viewpoint), while in the latter situation ∇ hasonly massless components.

21

3.1 The differential dϕ of ϕ and its decomposition induced by ∇

Three kinds of differentials, dϕ, Dϕ, and adϕ, of a map ϕ that naturally appear in the settingare defined and their local expressions are worked out in this subsection.

The differential, the covariant differential, and the inner differential of a map ϕ

Letϕ : (X,OAzX , E) −→ Y

be a map defined contravariantly by an equivalence class of gluing systems of ring-homomorphisms

ϕ] : OY −→ OAzX

over R ⊂ C. Then, for any derivation η on OAzX , the correspondence

OY −→ OAzXf 7−→ η(ϕ](f))

defines an OAzX -valued derivation on OY . It follows that ϕ induces a correspondence

T∗XAz −→ OAzX ⊗ϕ],OYT∗Yη 7−→ dηϕ

that is OCX -linear.

Definition 3.1.1. [differential dϕ of ϕ] The above OCX -linear correspondence is denoted by

dϕ and called the differential of ϕ.

Recall from Sec. 1 that when E is equipped with a connection ∇, ∇ induces a connection Don OAzX := EndO C

X(E), which in turn induces a splitting

T∗XC −→ T∗XAz

ξ 7−→ Dξ

of the exact sequence

0 −→ Inn (OAzX ) −→ T∗XAz −→ T∗XC −→ 0 .

Definition 3.1.2. [covariant differential Dϕ of ϕ] Let

ϕ∗T∗Y := OAzX ⊗ϕ],OY T∗Y ,

regarded as a (left) OX -module via the built-in inclusion OX → OAzX , be the pull-push of thetangent sheaf T∗Y of X to X. The covariant differential

Dϕ ∈ C∞(T ∗X ⊗OX ϕ∗T∗Y )

of ϕ is the (OAzX -valued-derivation-on-OY )-valued 1-form on X defined by

(Dξϕ)f := Dξ(ϕ](f)) ∈ C∞(End C(E))

for ξ ∈ C∞(T∗X) = Der (C∞(X)) and f ∈ C∞(Y ). In other words, Dϕ takes a tangent vectorfield on X to a C∞(End C(E))-valued derivation on C∞(Y ). In the equivalent sheaf format andnotations, Dξϕ ∈ ϕ∗T∗Y for ξ ∈ T∗X.

22

Definition 3.1.3. [inner differential adϕ of ϕ] Continuing Definition 3.1.2. Representelements in Inn (OAzX ) by elements m ∈ OAzX . The inner differential adϕ of ϕ is defined by theOCX -linear correspondence

adϕ : Inn (OAzX ) −→ ϕ∗T∗Ym 7−→ admϕ := [m, ϕ](· )] .

Since [OCX , · ] = 0, admϕ depends only on the inner derivation m represents.

By construction,dηϕ = Dξη + admηϕ ,

for η = ξη +mη ∈ T∗XAz ' T∗XC ⊕ Inn (OAzX ) induced by D.

Local expressions of the covariant differential Dϕ of ϕ for (∗1)-admissible (ϕ,∇)

Note that if ϕ : (XAz, E ;∇)→ Y is (∗1)-admissible, then Dξϕ is a Comm (Aϕ)-valued derivationon OY for ξ ∈ T∗X. In this case, one has the following lemma and corollary that are simplyre-writings of Lemma 2.1.6 and Lemma 2.1.7 respectively.

Lemma 3.1.4. [local expression of Dϕ for (∗1)-admissible (ϕ,∇), I] Let (ϕ,∇) be (∗1)-admissible; i.e. DξAϕ ⊂ Comm (Aϕ) for all ξ ∈ T∗X. Let U ⊂ X be a small enough open setso that ϕ(UAz) is contained in a coordinate chart of Y , with coordinates y = (y1, · · · , yn). Forf ∈ C∞(Y ), recall the germwise-over-U polynomial Rf [1] in (y1, · · · , yn) with coefficients inOAzU from [L-Y8: Sec. 4 & Remark/Notation 4.2.3.5] (D.13.1). Then, for ξ a vector field on Uand f ∈ C∞(Y ), and at the level of germs over U ,

(Dξϕ) f = Rf [1]∣∣yd Dξ(ϕ](yd)), for all multi-degree d in Rf [1]

.

Here, for a multiple degree d = (d1, , · · · , dn), di ∈ Z≥0, yd := (y1)d1 · · · (yn)dn andyd Dξ(ϕ

](yd)) means ‘replacing yd by Dξ(ϕ](yd))’.

Corollary 3.1.5. [local expression of Dϕ for (∗1)-admissible (ϕ,∇), II] Assume that(ϕ,∇) is (∗1)-admissible. Let dY be the exterior differential on Y . Then

Dϕ = ϕdY .

Locally explicitly, let (ei)i=1, ··· , n be a local frame on Y and (ei)i=1, ··· , n its dual co-frame. Interms of these dual pair of local frames, dY =

∑ni=1 e

i ⊗ ei under the canonical isomorphismT ∗Y ⊗OY OY ' T ∗Y . Then

(Dξϕ) f =

n∑i=1

(ϕei)(ξ)⊗ eif =

n∑i=1

(ϕei)(ξ)ϕ](eif) ∈ OAzX

under the canonical isomorphism OAzX ⊗ϕ],OY OY ' OAzX . In particular, let (y1, · · · , yn) be

coordinates of a local chart on Y . Then, locally,

(Dξϕ) f =

n∑i=1

(ϕdyi)(ξ)⊗ ∂∂yif =

n∑i=1

(Dξϕ

](yi) · ϕ]( ∂f∂yi

)).

Here · is the multiplication in OAzX (and will be omitted later when there is no sacrifice toclarity).

23

Local expressions of the inner differential adϕ of ϕ for admissible inner derivations

Though for the purpose of defining an action functional for D-branes, (ϕ,∇) is the dynamicalfield of the focus and, hence, the induced covariant derivation D on OAzX may look to playmore roles in our discussion, mathematically results related to D like Lemma 1.6, Lemma 3.1.4,and Corollary 3.1.5 involve only the fact that, for ξ ∈ TξX, Dξ satisfies the Leibniz rule onOAzX and some additional commutativity assumption. This suggest that similar statements holdif one considers inner derivations on OAzX that are compatible to ϕ in an appropriate sense.Cf. Lemma 1.6 vs. Lemma 1.7. This motivates the setting of the current theme.

Definition 3.1.6. [admissible inner derivation] (Cf. Definition 2.1.1.) Let m ∈ Inn (OAzX )be an inner derivation represented by an element of OAzX . Then m is called

(∗1)-admissible to ϕ if [m,Aϕ] ⊂ Comm (Aϕ) ;

(∗2)-admissible to ϕ if [m,Comm (Aϕ)] ⊂ Comm (Aϕ) ;

(∗3)-admissible to ϕ if [m,Aϕ] ⊂ Aϕ .

Note that these conditions are independent of the representative chosen in OAzX of the innerderivation. The set of all (∗1)-admissible-to-ϕ inner derivations on OAzX form an OC

X -module,which will be denoted by Inn ϕ

(∗1)(OAzX ). Similarly, for Inn ϕ

(∗2)(OAzX ) and Inn ϕ

(∗3)(OAzX ).

Lemma 3.1.7. [hierarchy of admissible conditions on inner derivation] (Cf. Lemma 2.1.2.)

Inn ϕ(∗3)(O

AzX ) ⊂ Inn ϕ

(∗2)(OAzX ) ⊂ Inn ϕ

(∗1)(OAzX ) .

Proof. Since Aϕ ⊂ Comm (Aϕ), it is immediate that Inn ϕ(∗2)(O

AzX ) ⊂ Inn ϕ

(∗1)(OAzX ). For the

inclusion Inn ϕ(∗3)(O

AzX ) ⊂ Inn ϕ

(∗2)(OAzX ), let m ∈ Inn ϕ

(∗3)(OAzX ) represented by an element in OAzX ,

m′ ∈ Comm (Aϕ), and m′′ ∈ Aϕ. Then,

[[m,m′],m′′] = [[m,m′′],m′] + [m, [m′,m′′]]

from either the Jacobi identity of Lie bracket or the Leibniz rule for a derivation. The first termvanishes since [m,m′′] ∈ Aϕ and [Aϕ,m′] = 0. The second term also vanishes since [m′,m′′] =0. Since m′ ∈ Comm (Aϕ) and m′′ ∈ Aϕ are arbitrary, this shows that [m,Comm (Aϕ)] ⊂Comm (Aϕ). This proves the lemma.

Remark 3.1.8. [Lemma 2.1.2 vs. Lemma 3.1.7 ] Note that in Lemma 2.1.2, the implication(∗3)⇒ (∗2) uses D-parallel transport properties implied by (∗3), which is an analytic technique.Indeed, the proof of Lemma 3.1.7 applies there. Which says that in both situations, the hierarchyis an algebraic consequence.

In terms of this setting and with arguments parallel to the proof of Lemma 3.1.4 and Corol-lary 3.1.5, one has the following lemma and corollary:

24

Lemma 3.1.9. [local expression of adϕ for Inn ϕ(∗1)(O

AzX ), I] (Cf. Lemma 3.1.4.)

Let ϕ : (XAz, E) → Y be a map and m ∈ Inn ϕ(∗1)(O

AzX ) represented by an element of OAzX i.e.

[m,Aϕ] ⊂ Comm (Aϕ). Let U ⊂ X be a small enough open set so that ϕ(UAz) is containedin a coordinate chart of Y , with coordinates y = (y1, · · · , yn). For f ∈ C∞(Y ), recall thegermwise-over-U polynomial Rf [1] in (y1, · · · , yn) with coefficients in OAzU from [L-Y8: Sec. 4& Remark/Notation 4.2.3.5] (D.13.1). Then, at the level of germs over U ,

(admϕ) f = Rf [1]∣∣yd adm(ϕ](yd)), for all multi-degree d in Rf [1]

.

Here, for a multiple degree d = (d1, , · · · , dn), di ∈ Z≥0, yd := (y1)d1 · · · (yn)dn andyd adm(ϕ](yd)) means ‘replacing yd by adm(ϕ](yd))’.

Proof. Recall the proof of Lemma 3.1.4 through the proof of Lemma 2.1.6. . With the samesetup and notations there, note that for m ∈ Inn ϕ

(∗1)(OAzX ),

[m,ϕ](1(k))] = 0 , for k = 1, · · · , s

by Lemma 1.7. Since [m,OX ] = 0 holds automatically, the lemma follows.

Corollary 3.1.10. [local expression of adϕ for Inn ϕ(∗1)(O

AzX ), II] (Cf. Corollary 3.1.5.)

Continuing the setting of Lemma 3.1.9. Recall that m ∈ Inn ϕ(∗1)(O

AzX ) is represented by an

element of OAzX . Let (y1, · · · , yn) be coordinates of a local chart on Y . Then, locally,

(admϕ) f =

n∑i=1

(admϕ

](yi) · ϕ]( ∂f∂yi

))

=n∑i=1

([m,ϕ](yi)] · ϕ]( ∂f

∂yi)),

where · is the multiplication in OAzX (and will be omitted later when there is no sacrifice toclarity). In other words,

admϕ =

n∑i=1

[m,ϕ](yi)]⊗ ∂∂yi .

Proof. The related last part of the proof of Corollary 3.1.5 through the proof of Lemma 2.1.7works in verbitum with Dξ replaced by adm = [m, · ]. This is simply a re-writing of Lemma 3.1.9above.

Remark 3.1.11. [comparison with differential of ordinary map] As a comparison, let u : X → Ybe a map between manifolds. Then, du defines a bundle map T∗X → u∗T∗Y that satisfies

du(ξ) f = u∗(ξ) f = ξ(u f) = ξ(u](f))

for any ξ ∈ T∗X and f ∈ C∞(Y ). In terms of local coordinates x= (xµ)µ=1, ··· ,m on X andy= (yi)i=1, ··· , n on Y ,

u∗

(∂∂xµ

)=

n∑i=1

dyi

dxµ∂f∂yi (u(x)) .

25

The above notion of covariant differential Dϕ of ϕ is exactly the generalization of the ordinarydifferential du of u, taking into account the fact that ϕ is now only defined contravariantlythrough ϕ] and that the noncommutative structure sheaf OAzX of XAz is no longer naturallytrivial as an OX -module but, rather, is endowed with a natural induced connection D from ∇.

Furthermore, when (ϕ,∇) is (∗1)-admissible or when considering only Inn ϕ(∗1)(O

AzX ), both the

covariant differential Dϕ and the inner differential adϕ takes the same form as the chain rulein the commutative case.

3.2 The three basic O CX-modules relevant to Dϕ, with induced structures

Underlying the notion of covariant differential Dϕ of ϕ are two basic OCX -modules

· the pull-back tangent sheaf ϕ∗T∗Y := OAzX ⊗ϕ],OYT∗Y , a left OAzY -module but now regarded

as a (left) OCX -module through the built-in inclusion OC

X → OAzX , and

· the OCX -module T ∗X ⊗OXϕ∗T∗Y , where Dϕ lives.

We study them in this subsection after taking a look at another basic but simpler OCX -module

T ∗X ⊗OXOAzX . They play fundamental roles in our variational problem.

Remark 3.2.0.1. [structures on the OCX-algebra OAzX ] Recall the connection D on the noncom-

mutative structure sheaf OAzX := EndO CX

(E) of XAz induced by the connection ∇ on E . The

multiplication · in the OCX -algebra structure of OAzX defines a nonsymmetric OAzX -valued inner

product on OAzX that is OCX -bilinear. This inner product is D-invariant in the sense that

D(m1 ·m2) = (Dm1) ·m2 + m1 ·Dm2 ,

for m1, m2 ∈ OAzX . Together with the built-in trace map

Tr ; : OAzX −→ OCX ,

as an OCX -module-homomorphism, one has a symmetric OC

X-valued inner product on OAzX definedby the assignment

(m1,m2) 7−→ Tr (m1 ·m2) =: Tr (m1m2) ,

for m1, m2 ∈ OAzX . This inner product is OCX -bilinear; and is covariantly constant over X in the

sense thatdX (Tr (m1m2)) = Tr ((Dm1)m2) + Tr (m1Dm2) ,

where dX is the exterior differential on X.

3.2.1 The OAzX -valued cotangent sheaf T ∗X ⊗OXOAzX of X, and beyond

Let X be endowed with a (Riemannian or Lorentzian) metric h and ∇h be the Levi-Civitaconnection on T∗X induced by h. The corresponding inner product on T∗X, its dual T ∗X, andtheir tensor products will be denoted 〈 · , · 〉h. The induced connection on the dual T ∗X and onthe tensor product of copies of T∗X and copies of T ∗X will be denoted also by ∇h. The definingfeatures of ∇h are

∇h〈ξ1, ξ2〉h = 〈∇hξ1, ξ2〉h + 〈ξ1,∇hξ2〉h (h be ∇h-covariantly constant) ,

Tor∇h(ξ1, ξ2) := ∇hξ1ξ2 −∇hξ2ξ1 − [ξ1, ξ2] = 0 (∇h be torsionless) ,

for all ξ1, ξ2 ∈ T∗X.

26

The OAzX -valued cotangent sheaf T ∗X ⊗OXOAzX of X

The connection ∇h on T ∗X and the connection D on OAzX together induce a connection

∇(h,D) := ∇h ⊗ IdOAzX+ Id T ∗X ⊗D

on T ∗X ⊗OXOAzX . The inner product 〈 · , · 〉h on T∗X and the inner product · on OAzX togetherinduce an OAzX -valued, OC

X-bilinear (nonsymmetric) inner product on T ∗X⊗OXOAzX by extendingOCX -bilinearly

〈ω1 ⊗m1 , ω2 ⊗m2〉h := 〈ω1 , ω2〉h · (m1Tm

2T ) ,

where ω1, ω2 ∈ T ∗X and m1, m2 ∈ OAzX . The trace map Tr : OAzX → OCX turns this further to

an OCX-valued, OC

X-bilinear (symmetric) inner product on T ∗X⊗OXOAzX by the post compositionwith the above inner product

( · , ·′) 7−→ Tr (〈 · , ·′〉h) =: Tr 〈 · , ·′〉hfor ·, ·′ ∈ T ∗X ⊗OXOAzX . By construction, both inner products are covariantly constant withrespect to ∇(h,D) and they satisfy the Leibniz rules

D〈 · , ·′〉h = 〈∇(h,D) · , ·′〉h + 〈 · , ∇(h,D) ·′〉g ,dTr 〈 · , ·′〉h = Tr (D〈 · , ·′〉h) = Tr 〈∇(h,D) · , ·′〉h + Tr 〈 · , ∇(h,D) ·′〉h

for ·, ·′ ∈ T ∗X ⊗OXOAzX .

The sheaf (∧• T ∗X)⊗OXOAzX of OAzX -valued differential forms on X

The setting in the previous theme generalizes to the sheaf (∧• T ∗X) ⊗OXOAzX of OAzX -valued

differential forms on X, with the 1-forms ω1, ω2 on X there replaced by general differentialforms α1, α2 on X. We will use the same notations

∇h , ∇(h,D) , 〈 · , · 〉h , Tr 〈 · , · 〉hto denote the connection on

∧• T ∗X, the connection on (∧• T ∗X)⊗OXOAzX , the OAzX -valued, OC

X-bilinear (nonsymmetric) inner product on (

∧• T ∗X)⊗OXOAzX , and the OCX-valued, OC

X-bilinear(symmetric) inner product on (

∧• T ∗X) ⊗OXOAzX respectively. They satisfy the same Leibnizrule as in the case of T ∗X ⊗OXOAzX

3.2.2 The pull-back tangent sheaf ϕ∗T∗Y

This is the main character among the three basic OCX -modules and is slightly subtler than u∗T∗Y

in the commutative case (cf. Remark 3.1.11) or T ∗X ⊗OXOAzX in Sec. 3.2.1.

The induced connection and the induced partially-defined inner products

Let Y be endowed with a (Riemannian or Lorentzian) metric g and ∇g be the Levi-Civitaconnection on T∗Y induced by g. The corresponding inner product on T∗Y or its dual T ∗Ywill be denoted 〈 · , · 〉g. The induced connection on the dual T ∗Y and on the tensor product ofcopies of T∗Y and copies of T ∗Y will be denoted also by ∇g. The defining features of ∇g are

∇g〈v1, v2〉g = 〈∇gv1, v2〉g + 〈v1,∇gv2〉g (g be ∇g-covariantly constant) ,

Tor∇g(v1, v2) := ∇gv1v2 −∇gv2v1 − [v1, v2] = 0 (∇g be torsionless) ,

for all v1, v2 ∈ T∗Y .

27

Lemma 3.2.2.1. [(D,∇g)-induced connection on ϕ∗T∗Y for (∗)1-admissible (ϕ,∇)] As-sume that (ϕ,∇) is (∗1)-admissible. Then, the connection D on OAzX and the connection ∇g onT∗Y together induce a connection ∇(ϕ,g) on ϕ∗T∗Y , locally of the form

∇(ϕ,g) = D ⊗ Id T∗Y + IdOAzX·n∑i=1

Dϕ](yi)⊗∇g∂∂yi

.

Proof. Our construction of an induced connection on ϕ∗T∗Y is local in nature. As long asa construction is independent of coordinates chosen, the local construction glues to a globalconstruction. Let U ⊂ X be a small enough open set so that ϕ(UAz) is contained in a coordinatechart of Y , with coordinate y := (y1, · · · , yn). Then the local expression

∇(ϕ,g);y := D ⊗ Id T∗Y + IdOAzX·n∑i=1

Dϕ](yi)⊗∇g∂∂yi

.

in the Statement defines a connection on ϕ∗T∗Y |U . We only need to show that it is independentof the coordinate (y1, · · · , yn) chosen.

Let z := (z1, · · · , zn) be another coordinate on the chart. Then, for (ϕ,∇) (∗1)-admissible,Dϕ](yi) =: (Dϕ)yi has a local expression in terms of z

(Dϕ)yi := D(ϕ](yi)) =n∑j=1

D(ϕ](zj))⊗ ∂yi

∂zj

by Corollary 3.1.5, for i = 1, · · · , n. It follows that

∇(ϕ,g);z := D ⊗ Id T∗Y + IdOAzX·n∑j=1

Dϕ](zj)⊗∇g∂∂zj

= D ⊗ Id T∗Y + IdOAzX·n∑j=1

Dϕ](zj)⊗n∑i=1

∂yi

∂zj∇g∂

∂yi

= D ⊗ Id T∗Y + IdOAzX·n∑i=1

Dϕ](yi)⊗∇g∂∂yi

=: ∇(ϕ,g);y .

This completes the proof.

Consider next the induced inner products on ϕ∗T∗Y . Completely naturally, one may attempto combine the multiplication in OAzX and the inner product 〈 · , · 〉g on T∗Y to an OAzX -valued,OCX -bilinear (nonsymmetric) inner product on ϕ∗T∗Y by extending OC

X -bilinearly

〈m1 ⊗ v1 , m2 ⊗ v2〉g := m1m2 ⊗ 〈v1 , v2〉g = m1m2ϕ](〈v1 , v2〉g) ,

where m1, m2 ∈ Comm (Aϕ) ⊂ OAzX and v1, v2 ∈ T∗Y and the last equality follows from thecanonical isomorphism

OAzX ⊗ϕ],OYOY ' OAzX .

However, for this to be well-defined, it is required that

〈m1 ⊗ f1v1 , m2 ⊗ f2v2〉g = 〈m1ϕ](f1)⊗ v1 , m

2ϕ](f2)⊗ v2〉g ,

28

i.e.m1m2ϕ]

(f1f2〈v1, v2〉g

)= m1ϕ](f1)m2ϕ](f2)ϕ](〈v1, v2〉g) ,

for all m1,m2 ∈ OAzX , v1, v2 ∈ T∗Y , and f1, f2 ∈ OY . Which holds if and only if

m2 ∈ Comm (Aϕ) .

What happens if one brings in the trace map Tr : OAzX → OCX ? In this case,

Tr 〈m1 ⊗ f1v1 , m2 ⊗ f2v2〉g = Tr

(m1m2ϕ](f1f2〈v1, v2〉g)

)= Tr

(ϕ](f1)m1m2ϕ](f2〈v1, v2〉g)

)by the cyclic-invariance property of Tr while

Tr 〈m1ϕ](f1)⊗ v1 , m2ϕ](f2)⊗ v2〉g = Tr

(m1ϕ](f1)m2ϕ](f2〈v1, v2〉g)

).

The two equal if

either m1 ∈ Comm (Aϕ) or m2 ∈ Comm (Aϕ) .

Definition 3.2.2.2. [partially-defined inner product 〈 · , · 〉g on ϕ∗T∗Y ] The multiplica-tion in OAzX and the inner product 〈 · , · 〉g on T∗Y together induce a partially defined, OAzX -valued,OCX-bilinear (nonsymmetric) inner product on ϕ∗T∗Y by extending OC

X -bilinearly

〈m1 ⊗ v1 , m2 ⊗ v2〉g := m1m2 ⊗ 〈v1 , v2〉g = m1m2ϕ](〈v1 , v2〉g) ,

where m1 ∈ OAzX , m2 ∈ Comm (Aϕ) ⊂ OAzX and v1, v2 ∈ T∗Y and the last equality follows fromthe canonical isomorphism OAzX ⊗ϕ],OYOY ' O

AzX .

Definition 3.2.2.3. [partially-defined inner product Tr 〈 · , · 〉g on ϕ∗T∗Y ] The multipli-cation in OAzX , the inner product 〈 · , · 〉g on T∗Y , and the tarce map Tr : OAzX → OC

X togetherinduce a partially defined, OC

X-valued, OCX-bilinear (symmetric) inner product on ϕ∗T∗Y by ex-

tending OCX -bilinearly

Tr 〈m1 ⊗ v1 , m2 ⊗ v2〉g := Tr

(m1m2 ⊗ 〈v1 , v2〉g

)= Tr

(m1m2ϕ](〈v1 , v2〉g)

),

where either m1 or m2 is in Comm (Aϕ), v1, v2 ∈ T∗Y , and the last equality follows from thecanonical isomorphism OAzX ⊗ϕ],OYOY ' O

AzX .

By construction, both inner products, when defined, are covariantly constant with respect to∇(ϕ,g) and one has the Leibniz rules

D〈 – , –′〉g = 〈∇(ϕ,g) – , –′〉g + 〈 – , ∇(ϕ,g) –′〉g ,dXTr 〈 – , –′〉g = Tr (D〈 – , –′〉g) = Tr 〈∇(ϕ,g) – , –′〉g + Tr 〈 – , ∇(ϕ,g) –′〉g ,

whenever the 〈 –′′ , –′′′〉g or Tr 〈 –′′ , –′′′〉g involved are defined.The followng lemma is an immediate consequence of Corollary 3.1.5:

29

Lemma 3.2.2.4. [sample list of defined inner products for admissible (ϕ,∇)] (1) For(ϕ,∇) (∗1)-admissible, the following list of inner products

〈 – , Dξϕ〉g , Tr 〈 – , Dξϕ〉g , Tr 〈Dξϕ , – 〉g

are defined for ξ ∈ T∗X.(2) For (ϕ,∇) (∗2)-admissible, the following additional list of inner products

〈 – ,∇(ϕ,g)ξ1

· · · ∇(ϕ,g)ξk

Dξϕ〉g , Tr 〈 – ,∇(ϕ,g)ξ1

· · · ∇(ϕ,g)ξk

Dξϕ〉g , Tr 〈∇(ϕ,g)ξ1

· · · ∇(ϕ,g)ξk

Dξϕ , , – 〉g

are also defined for ξ, ξ1, · · · , ξk ∈ T∗X, k ∈ Z≥0.

The symmetry properties of the curvature tensor of ∇(ϕ,g) on ϕ∗T∗Y

Let ϕ : (XAz, E ;∇) → Y be a (∗2)-admissible map. Let F∇(ϕ,g) be the curvature tensor of theinduced connection ∇(ϕ,g) on ϕ∗T∗Y — the EndO C

X(ϕ∗T∗Y )-valued 2-form on X defined by

F∇(ϕ,g)(ξ1, ξ2) s :=(∇(ϕ,g)ξ1∇(ϕ,g)ξ2

− ∇(ϕ,g)ξ2∇(ϕ,g)ξ1

− ∇(ϕ,g)[ξ1,ξ2]

)s ∈ ϕ∗T∗Y

for ξ1, ξ2 ∈ T∗X and s ∈ ϕ∗T∗Y . (This is OX -linear in ξ1, ξ2, and s and, hence, a tensor on X).By construction,

F∇(ϕ,g)(ξ1, ξ2) = −F∇(ϕ,g)(ξ2, ξ1)

for ξ1, ξ2 ∈ T∗X. From Lemma 3.2.2.4, the inner products

〈∇(ϕ,g)ξ1

Dξ3ϕ , Dξ4ϕ〉g , 〈∇(ϕ,g)ξ1

Dξ3 , ∇(ϕ,g)ξ2

Dξ4ϕ〉g ,

〈∇(ϕ,g)ξ1∇(ϕ,g)ξ2

Dξ3ϕ , Dξ4ϕ〉g , 〈F∇(ϕ,g)(ξ1, ξ2)Dξ3ϕ , Dξ4ϕ〉g

are all defined.

Lemma 3.2.2.5. [symmetry property of the curvature tensor of ∇(ϕ,g) on ϕ∗T∗Y ] Fora (∗2)-admissible pair (ϕ,∇),

〈F∇(ϕ,g)(ξ1, ξ2)Dξ3ϕ , Dξ4ϕ〉g= − 〈Dξ3ϕ , F∇(ϕ,g)(ξ1, ξ2)Dξ4ϕ〉g + [F∇(ξ1, ξ2) , 〈Dξ3ϕ , Dξ4ϕ〉g] .

And, hence,

Tr 〈F∇(ϕ,g)(ξ1, ξ2)Dξ3ϕ , Dξ4ϕ〉g = −Tr 〈Dξ3ϕ , F∇(ϕ,g)(ξ1, ξ2)Dξ4ϕ〉g= −Tr 〈F∇(ϕ,g)(ξ1, ξ2)Dξ4ϕ , Dξ3ϕ〉g = Tr 〈F∇(ϕ,g)(ξ2, ξ1)Dξ4ϕ , Dξ3ϕ〉g .

Proof. This is a consequence of the Leibniz rule

D〈 – , –′〉g = 〈∇(ϕ,g) – , –′〉g + 〈 – , ∇(ϕ,g) –′〉g ,

30

for –, –′ ∈ ϕ∗T∗Y , when all inner products involved are defined. In detail,

〈F∇(ϕ,g)(ξ1, ξ2)Dξ3ϕ , Dξ4ϕ〉g

= 〈(∇(ϕ,g)ξ1∇(ϕ,g)ξ2−∇(ϕ,g)

ξ2∇(ϕ,g)ξ1−∇(ϕ,g)

[ξ1,ξ2])Dξ3ϕ , Dξ4ϕ〉g

= Dξ1〈∇(ϕ,g)ξ2

Dξ3ϕ , Dξ4ϕ〉g − 〈∇(ϕ,g)ξ2

Dξ3ϕ , ∇(ϕ,g)ξ1

Dξ4ϕ〉g

− Dξ2〈∇(ϕ,g)ξ1

Dξ3ϕ , Dξ4ϕ〉g + 〈∇(ϕ,g)ξ1

Dξ3ϕ , ∇(ϕ,g)ξ2

Dξ4ϕ〉g

− D[ξ1,ξ2]〈Dξ3ϕ , Dξ4ϕ〉g + 〈Dξ3ϕ , ∇(ϕ,g)[ξ1,ξ2]Dξ4ϕ〉g

= 〈Dξ3ϕ , (∇(ϕ,g)ξ2∇(ϕ,g)ξ1−∇(ϕ,g)

ξ1∇(ϕ,g)ξ2

+∇(ϕ,g)[ξ1,ξ2])Dξ4ϕ〉g

+ (Dξ1Dξ2 −Dξ2Dξ1 −D[ξ1,ξ2])〈Dξ3ϕ , Dξ4ϕ〉gafter repeatedly applying the Leibniz rule,

= − 〈Dξ3ϕ , F∇(ϕ,g)(ξ1, ξ2)Dξ4ϕ〉g + FD(ξ1, ξ2)〈Dξ3ϕ , Dξ4ϕ〉g .

Note that FD = [F∇, · ] on OAzX . The lemma follows.

Covariant differentiation and evaluation

Lemma 3.2.2.6. [Dξ(m⊗vf) vs.(∇(ϕ,g)ξ (m⊗v)

)f ] Let ξ ∈ T∗X, f ∈ OY , and m⊗v ∈ ϕ∗T∗Y .

Then,

Dξ(m⊗ vf) =(∇(ϕ,g)ξ (m⊗ v)

)f + m

n∑i=1

Dξϕ](yi)⊗

(∂∂yi (vf)−

(∇g∂

∂yi

v)f).

Proof.Dξ(m⊗ vf) = Dξ

(mϕ](vf)

)= Dξm⊗ vf +

∑i

Dξϕ](yi)⊗ ∂

∂yi(vf)

while (∇(ϕ,g)ξ (m⊗ v)

)f =

(Dξm⊗ v +

∑i

Dξϕ](yi)⊗∇g∂

∂yi

v)f .

The lemma follows.

3.2.3 The OCX-module T ∗X ⊗OX ϕ∗T∗Y , where Dϕ lives

Assume that (ϕ,∇) is (∗1)-admissible and recall the metric h on X and the metric g on Y .Then the construction in this subsubsection is a combination of the constructions in Sec. 3.2.1and Sec. 3.2.2.

The connection ∇h on T ∗X, the connection D on OAzX , and the connection ∇g on T∗Ytogether induce a connection ∇(h,ϕ,g) on T ∗X ⊗OXϕ∗T∗Y , locally of the form

∇(h,ϕ,g) = ∇h ⊗ IdOAzX ⊗ Id T∗Y + Id T ∗X ⊗DT ⊗ Id T∗Y + Id T ∗X ⊗ IdOAzX ·n∑i=1

Dϕ](yi)⊗∇g∂∂yi

.

31

Lemma 3.2.2.1 implies that this is independent of the coordinate (y1, · · · , yn) on coordinatecharts chosen and hence well-defined

The inner product 〈 · , · 〉h on T ∗X, the multiplication inOAzX , and the inner product 〈 · , · 〉g onT∗Y together induce a partially defined, OAzX -valued, OC

X-bilinear (nonsymmetric) inner producton T ∗X ⊗OXϕ∗TT∗Y by extending OC

X -bilinearly

〈ω1 ⊗m1T ⊗ v1 , ω

2m2 ⊗ v2〉(h,g):= 〈ω1 , ω2〉h ·m1

Tm2T ⊗ 〈v1 , v2〉g = 〈ω1 , ω2〉hm1m2 ϕ](〈v1 , v2〉g) ,

where ω1, ω2 ∈ T ∗X, m1 ∈ OAzX , m2 ∈ Comm (Aϕ) ⊂ OAzX , v1, v2 ∈ T∗Y and the last equalityfollows from the canonical isomorphism OAzX ⊗ϕ],OYOY ' O

AzX . The trace map Tr : OAzX → OC

X

gives another partially defined, OCX-valued, OC

X-bilinear (symmetric) inner product on T ∗X⊗OXϕ∗T∗Y by extending OC

X -bilinearly

Tr 〈ω1 ⊗m1T ⊗ v1 , ω

2m2 ⊗ v2〉(h,g):= Tr

(〈ω1 , ω2〉h ·m1

Tm2T ⊗ 〈v1 , v2〉g

)= Tr

(〈ω1 , ω2〉hm1m2 ϕ](〈v1 , v2〉g)

),

where ω1, ω2 ∈ T ∗X, either m1 or m2 is in Comm (Aϕ), v1, v2 ∈ T∗Y .By construction, both inner products, when defined, are covariantly constant with respect to

∇(h,ϕ,g) and one has the Leibniz rules

D〈∼ , ∼′〉(h,g) = 〈∇(h,ϕ,g) ∼ , ∼′〉(h,g) + 〈∼ , ∇(h,ϕ,g) ∼′〉(h,g) ,

dTr 〈∼ , ∼′〉(h,g) = Tr (D〈∼ , ∼′〉(h,g))

= Tr 〈∇(h,ϕ,g) ∼ , ∼′〉(h,g) + Tr 〈∼ , ∇(h,ϕ,g) ∼′〉(h,g) ,

whenever the 〈∼′′ , ∼′′′〉(h,g) or Tr 〈∼′′ , ∼′′′〉(h,g) involved are defined.

With all the preparations in Sec. 1–Sec. 3, we are finally ready to construct and study thestandard action for D-branes along our line of pursuit.

4 The standard action for D-branes

We introduce in this section the standard action, which is to D-branes as the (Brink-Di Vecchia-Howe/Deser-Zumino/)Polyakov action is to fundamental strings. Abstractly, it is an enhancednon-Abelian gauged sigma model based on maps ϕ : (XAz, E ;∇)→ Y .

The gauge-symmetry group C∞(Aut C(E))

Let Aut C(E) be the automorphism bundle of the complex vector bundle E (of rank r) over E.Aut C(E) ⊂ End C(E) canonically as the bundle of invertible endomorphisms; it is a principalGLr(C)-bundle over X. The set

Ggauge := C∞(Aut C(E))

of smooth sections of Aut C(E) forms an infinite-dimensional Lie group and acts on the space ofpairs (ϕ,∇) as a gauge-symmetry group:

g′ ∈ Ggauge : (ϕ,∇ = d+A) 7−→ (g′ϕ , g

′∇ = d+ g′A)

:=(g′ϕg′−1 , d− (dg′)g′−1 + g′Ag′−1) .

The induced action of Ggauge on other basic objects are listed in the lemma below:

32

Lemma 4.1. [induced action of Ggauge on other basic objects] (All the Ggauge -actionsare denoted by a representation ρgauge of Ggauge , if in need.)

(01) on OAzX : ρgauge (g′)(m) = g′mg′−1 for m ∈ OAzX .

(02) on induced connections : D = d+ [A, · ] 7−→ g′D := d+ [ g′A , · ].

(1) on T ∗X C ⊗O CXOAzX : ρgauge (g′)(ω ⊗m) = ω ⊗ (g′mg′−1) =: g′(ω ⊗m)g′−1.

(2) for ϕ∗T∗Y :

ϕ∗T∗Y −→ g′ϕ∗T∗Ym⊗ v 7−→ (g′mg′−1)⊗ v =: g′(m⊗ v)g′−1 .

.

(3) for T ∗X ⊗OXϕ∗T∗Y :

T ∗X ⊗OXϕ∗T∗Y −→ T ∗X ⊗OX g′ϕ∗T∗Y

ω ⊗m⊗ v 7−→ ω ⊗ (g′mg′−1)⊗ v =: g′(ω ⊗m⊗ v)g′−1 .

.

(4) for covariant differential : Dϕ 7−→ g′D g′ϕ = g′Dϕg′−1.

(5) for pull-push : (g′ϕ)α = g′ ϕα g′−1.

Proof. The proof is elementary. Let us demonstrate Item (2) as an example.For m⊗ v ∈ ϕ∗T∗Y := OAzX ⊗ϕ],OYT∗Y ,

ρgauge (g′)(m⊗ v) = ρgauge (g′)(m)⊗ v = (g′mg′−1

)⊗ v

since Ggauge acts on T∗Y trivially (i.e. by by the identity map Id Y ). The only issue is: Where

does (g′mg′−1)⊗ v now live? To answer this, note that, for f ∈ C∞(Y ), on one hand

ρgauge (g′)(m⊗ fv) = ρgauge (g′)(mϕ](f)⊗ v) = (g′mϕ](f)g′−1

)⊗ v ,

while on the other hand

ρgauge (g′)(m⊗ fv) = (g′mg′−1

)⊗ fv ,

It follows that

(g′mg′−1

)⊗ fv

=(g′mϕ](f)g′

−1)⊗ v =(g′mg′

−1 · g′ϕ](f)g′−1)⊗ v =

(g′mg′

−1 · g′ϕ](f))⊗ v .

Which says that our section (g′mg′−1)⊗ v now lives in g′ϕ∗T∗Y .

33

The standard action for D-branes

Fix a (dilaton field ρ, metric h) on the underlying smooth manifold X (of dimension m) of theAzumaya/matrix manifold with a fundamental module (XAz, E). Fix a background (dilaton fieldΦ, metric g , B-field B, Ramond-Ramond field C) on the target space(-time) Y (of dimensionn).2 Here, h and g can be either Riemannian or Lorentzian.

Definition 4.2. [standard action = enhanced non-Abelian gauged sigma model]With the given background fields (ρ, h) on X and (Φ, g, B,C) on Y , the standard action

S(ρ,h;Φ,g,B,C)standard (ϕ,∇) for (∗1)-admissible pairs (ϕ,∇) is defined to be the functional

S(ρ,h;Φ,g,B,C)standard (ϕ,∇) := S

(ρ,h;Φ,g,B,C)

nAGSM+ (ϕ,∇)

:= S(ρ,h;Φ,g)

map:kinetic+(ϕ,∇) + S(h;B)gauge /YM(ϕ,∇) + S

(C,B)CS/WZ (ϕ,∇)

with the enhanced kinetic term for maps

S(ρ,h;Φ,g)

map:kinetic+(ϕ,∇) :=12 Tm−1

∫X

Re(

Tr 〈Dϕ , Dϕ〉(h,g))

vol h +

∫X

Re(

Tr 〈dρ, ϕdΦ〉h)

vol h ,

the gauge/Yang-Mills term

S(h;B)gauge /YM(ϕ,∇) := − 1

2

∫X

Re(

Tr ‖2πα′F∇ + ϕB‖2h)

vol h

and the Chern-Simons/Wess-Zumino term(if (ϕ,∇) is furthermore (∗2)-admissible, cf. Remark 2.1.13)

S(C,B)

CS/WZ(ϕ,∇)

formally= Tm−1

∫X

Re(

Tr(ϕC ∧ e2πα′F∇+ϕB ∧

√A(XAz)/A(NXAz/Y )

))(m)

.

Here,

(0) On Re Note that while eigenvalues of ϕ](f) are all real ([L-Y5: Sec. 3.1] (D(11.1)))for f ∈ OY , the eigenvalues of Dξϕ

](f), ξ ∈ T∗X, may not be so under the (∗1)-Admissible

Condition. Thus, Tr (· · · · · · ) in the integrand of terms in S(ρ,h;Φ,g,B,C)standard (ϕ,∇) are in general

C-valued and we take the real part Re Tr (· · · · · · ) of it.

(1) The enhanced kinetic term for maps The first summand of S(ρ,h;Φ,g)

map:kinetic+ defines the

kinetic energy

E∇(ϕ) := S(h;g)map:kinetic(ϕ,∇) :=

12 Tm−1

∫X

Re(

Tr 〈Dϕ , Dϕ〉(h,g))

vol h

of the map ϕ for a given ∇ and, hence, will be called the kinetic term for maps in the

standard action S(ρ,h;Φ,g,B,C)standard (ϕ,∇). When the metric g on Y is Lorentzian, then depending

on the convention of its signature (−,+, · · · +) vs. (+,−, · · · −), one needs to add anoverall minus − vs. plus + sign. In this note, for simplicity of presentation, we choose hand g to be both Riemannian (i.e. for Euclideanized/Wick-rotated D-branes and space-time).

2For mathematicians, ρ is a smooth function on X, Φ is a smooth function on Y , B is a 2-form and C isa general differential form on Y . Such background fields (Φ, g, B,C) on Y are created by massless excitationsof closed superstrings on Y . The notations for these particular fields are almost already carved into stone instring-theory literature. Which we adopt here.

34

· The world-volume XAz of D-brane is m-dimensional; Tm−1 is the tension of (m − 1)-dimensional D-branes. Like the tension of the fundamental string, it is a fixed constant ofnature.

· The second summand of S(ρ,h;Φ,g)

map:kinetic+

S(ρ,h;Φ)dilaton (ϕ) :=

∫X

Re(

Tr 〈dρ, ϕdΦ〉h)

vol h ,

will be called the dilaton term of the standard action S(ρ,h;Φ,g,B,C)standard (ϕ,∇).

Note that if let U be small enough and fix a local trivialization of E|U . and assumethat ∇ = d+A with respect to this local trivialization. Then D = d+ [A, · ] and, over Uwith an orthonormal frame (eµ)µ,

Tr 〈dρ, ϕdΦ〉h =∑µ

Tr(dρ(eµ)Deµϕ

](Φ))

=∑µ

Tr(dρ(eµ)

(eµϕ

](Φ) + [A(eµ), ϕ](Φ)]))

=∑µ

Tr(dρ(eµ)

(eµϕ

](Φ))).

Thus, while ϕdΦ depends on the connection ∇, the integrand(Tr 〈dρ, ϕdΦ〉h

)vol h does

not. This justifies the dilaton term as a functional of ϕ alone.

In contrast, over U with the above setting, Tr 〈Dϕ,Dϕ〉(h,g) contains summand∑µ

∑i,j

Tr(

[A(eµ), ϕ](yi)] [A(eµ), ϕ](yj)]ϕ](gij)),

which does not vanish in general. Thus, Tr 〈Dϕ,Dϕ〉(h,g) does depend on the pair (ϕ,∇).

(2) The gauge/Yang-Mills term S(h;B)gauge /YM(ϕ,∇) α′ is the Regge slope; 2πα′ is the inverse

to the tension of a fundamental string.

· F∇ is the curvature tensor of the connection ∇ on E; 2πα′F∇ + ϕB is an OAzX -valued2-tensor on X; and

‖2πα′F∇ + ϕB‖2h := 〈2πα′F∇ + ϕB , 2πα′F∇ + ϕB〉h

from Sec. 3.2.1. Up to the shift by ϕB, this is a norm-squared of the field strength of

the gauge field, and hence the name Yang-Mills term. Note that in S(h;B)gauge /YM(ϕ,∇), ∇

couples with ϕ only through the background B-field B. When B = 0, this is simply a

functional S(h)gauge /YM(∇) of ∇ alone.

· In the current bosonic case, the Yang-Mills functional for the gauge term S(h;B)gauge /YM(ϕ,∇)

can be replaced any other standard action functional, e.g. Chern-Simons functional, ingauge theories.

(3) The Chern-Simons/Wess-Zumino term S(C,B)CS/WZ (ϕ,∇) The coupling constant of

Ramond-Ramond fields with D-branes is taken to be equal to the D-brane tension Tm−1.This choice is adopted from the situation of the Dirac-Born-Infeld action. However, in

the current bosonic case, one may take a different constant. As given here, S(C,B)CS/WZ (ϕ,∇)

is only formal; the anomaly factor√A(XAz)/A(NXAz/Y ) in its integrand remains to be

understood in the current situation.

35

· The wedge product of OAzX -valued differential forms was discussed in [L-Y8: Sec.6.1](D(13.1)). An Ansatz was proposed there in accordance with the notion of ‘symmetrizeddeterminant’ for an OAzX -valued 2-tensor on X in the construction of the non-AbelianDirac-Born-Infeld action ibidem. Here, we no longer have a direct guide from the con-

struction of the kinetic term S(h;g)map:kinetic(ϕ,∇) for maps as to how to define such wedge

products. However, just like Polyakov string should be thought of as being equivalent toNambu-Goto string (at least at the classical level) but technically more robust, here wewould think that ‘standard D-branes’ should be equivalent to ‘Dirac-Born-Infeld D-branes’(at least classically) and, hence, will take the same Ansatz:

Ansatz [wedge product in the Chern-Simons/Wess-Zumino action] We in-terpret the wedge products that appear in the formal expression for the Chern-

Simons/Wess-Zumino term S(C,B)CS/WZ (ϕ,∇) through the symmetrized determinant that

applies to the above defining identities for wedge product; namely, we require that

(ω1 ∧ · · · ∧ ωs)(e1 ∧ · · · ∧ es) = SymDet (ωi(ej))

for OAzX -valued 1-forms ω1, · · · , ωs on X. Denote this generalized wedge product by∧.

Then, for lower-dimensional D-branes m = 0, 1, 2, 3, it is reasonable to assume that the

anomaly factor is 1 (i.e. no anomaly) and S(C,B)CS/WZ (ϕ,∇) can be written out precisely.

Locally in terms of a local frame (eµ)µ on an open set U ⊂ X and a coordinate(y1, · · · , yn) on a local chart of Y , one has: (Assuming that B =

∑i,j Bijdy

i ⊗ dyj ,Bji = −Bij .)

· For D(−1)-brane world-point (m = 0) :

S(C(0))CS/WZ (ϕ) = T−1 · Tr (ϕC(0)) = T−1 · Tr (ϕ](C(0))) .

· For D-particle world-line (m = 1) : Assume that C(1) =∑n

i=1Ci dyi locally; then

S(C(1))CS/WZ (ϕ) = T0

∫X

Re(Tr (ϕC(1))

) locally= T0

∫U

Re(

Tr( n∑i=1

ϕ](Ci) ·De1ϕ](yi)

))de1 .

Note that as in the case of the dilaton term S(ρ,h;Φ)dilaton (ϕ), this is a functional of ϕ alone.

· For D-string world-sheet (m = 2) : Assume that C(2) =∑n

i,j=1Cij dyi ⊗ dyj locally,

with Cij = −Cji; then

S(C(0),C(2),B)CS/WZ (ϕ,∇) = T1

∫X

Re (Tr (ϕC(2) + ϕ(C(0)B) + 2πα′ϕ](C(0)) F∇))

= T1

∫X

Re (Tr (ϕ(C(2) + C(0)B) + πα′ϕ](C(0))F∇ + πα′F∇ϕ](C(0))))

locally= T1

∫U

Re(

Tr( n∑i,j=1

ϕ](Cij + C(0)Bij)De1ϕ](yi)De2ϕ

](yj)

+πα′ϕ](C(0))F∇(e1, e2) + πα′F∇(e1, e2)ϕ](C(0))))

e1 ∧ e2

= T1

∫U

Re(

Tr( n∑i,j=1

ϕ](Cij + C(0)Bij)De1ϕ](yi)De2ϕ

](yj)

+ 2πα′ϕ](C(0))F∇(e1, e2)))

e1 ∧ e2 .

Here, the last identity comes from the effect of the trace map Tr .

36

· For D-membrane world-volume (m = 3) : Assume that C(1) =∑n

i=1Ci dyi and C(3) =∑n

i,j,k=1Cijk dyi ⊗ dyj ⊗ dyk locally, with Cijk alternating with respect to ijk; then

S(C(1),C(3),B)CS/WZ (ϕ,∇) = T2

∫X

Re (Tr (ϕC(3) + ϕ(C(1) ∧B) + 2πα′ ϕC(1)

∧ F∇ ))

locally= T2

∫U

Re(

Tr( n∑i,j,k=1

ϕ](Cijk + CiBjk + CjBki + CkBij)

·De1ϕ](yi)De2ϕ

](yj)De3ϕ](yk)

+πα′∑

(λµν)∈Sym3

n∑i=1

(−1)(λµν)(ϕ](Ci)Deλ(ϕ](yi))F∇(eµ, eν)

+F∇(eµ, eν)ϕ](Ci)Deλϕ](yi)

)))e1 ∧ e2 ∧ e3

= T2

∫U

Re(

Tr( n∑i,j,k=1

ϕ](Cijk + CiBjk + CjBki + CkBij)

·De1ϕ](yi)De2ϕ

](yj)De3ϕ](yk)

+ 2πα′∑

(λµν)∈Sym3

n∑i=1

(−1)(λµν)(ϕ](Ci)Deλ(ϕ](yi))F∇(eµ, eν)

)))e1 ∧ e2 ∧ e3 .

Here, the last identity comes from the effect of the trace map Tr .

Their partial study was done in [L-Y8 : Sec. 6.2] (D(13.1)).

(4) The background B-field The coupling of (ϕ,∇) with the background B-field B on Yin the part

S(h;B)gauge /YM(ϕ,∇) + S

(C,B)CS/WZ (ϕ,∇)

of the standard action means that we have to adjust the fundamental module E on Xby a compatible “twisting” governed by ϕ and B. With this “twisting”, E now lives ona gerb over X. See [L-Y2] (D(5)) for details and further references. However, since thestudy of the variational problems in this note is mainly local and focuses on the enhanced

kinetic term for maps S(ρ,h;Φ,g)

map:kinetic+ , we’ll ignore this twisting for the current note to keep

the language and expressions simple.

Remark 4.3. [other effects from B-field and Ramond-Ramond field ] There are other effects toD-branes beyond just mentioned above from the background B-field and Ramond-Ramond fieldthat have not yet been taken into account in this project so far; e.g. [H-M1], [H-M2], and [H-Y].They can influence the action for D-branes as well. Such additional effects should be investigatedin the future.

Theorem 4.4. [well-defined gauge-symmetry-invariant action] Except the anomaly fac-tor in the Chen-Simons/Wess-Zumino term, which is yet to be understood, the standard action

S(ρ,h;Φ,g,B,C)standard (ϕ,∇) as given in Definition 4.2 for (∗1)-admissible pairs (ϕ,∇) (and S

(C,B)CS/WZ (ϕ,∇)

for (∗2)-admissible (ϕ,∇)) is well-defined. Assume that the anomaly factor in the Chen-Simons/Wess-Zumino term transforms also by conjugation as for OAzX under a gauge symmetry, then

S(ρ,h;Φ,g,B,C)standard (ϕ,∇) is invariant under gauge symmetries:

S(ρ,h;Φ,g,B,C)standard (ϕ,∇) = S

(ρ,h;Φ,g,B,C)standard ( g

′ϕ, g

′∇)

for g′ ∈ Ggauge := C∞(Aut C(E)).

37

Proof. For the kinetic term for maps

S(h;g)map:kinetic(ϕ,∇) :=

12 Tm−1

∫X

Re(

Tr 〈Dϕ , Dϕ〉(h,g))

vol h ,

that it is well-defined follows Lemma 3.2.2.4. Under a gauge transformation g′ ∈ Ggauge :=C∞(Aut C(E)) and in terms of local coordinates (x1, · · · , xm) on X and (y1, · · · , yn) on Y ,

g′D g′ϕ =∑µ

dxµ ⊗∑i

g′D ∂∂xµ

g′ϕ]( ∂∂yi)⊗g′ϕ

∂∂yi

=∑µ

dxµ ⊗∑i

(g′(D ∂

∂xµϕ]( ∂∂yi))g′−1)⊗g′ϕ

∂∂yi

.

Thus,

〈 g′D g′ϕ , g

′D g′ϕ〉(h,g)

=∑µ,ν

∑i,j

hµν ⊗(g′(D ∂

∂xµϕ]( ∂∂yi))g′−1 · g′

(D ∂

∂xνϕ]( ∂∂yj))g′−1)⊗g′ϕ gij

=∑µ,ν

∑i,j

hµν ·(g′(D ∂

∂xµϕ]( ∂∂yi))g′−1 · g′

(D ∂

∂xνϕ]( ∂∂yj))g′−1)· g′ϕ](gij)g′

−1

= g′(∑µ,ν

∑i,j

hµν ·D ∂∂xµ

ϕ]( ∂∂yi)·D ∂

∂xνϕ]( ∂∂yj)· ϕ](gij)

)g′−1

= g′ 〈Dϕ , Dϕ〉(h,g) g′−1.

It follows that Tr 〈 g′D g′ϕ , g′D g′ϕ〉(h,g) = Tr 〈Dϕ , Dϕ〉(h,g) and, hence,

S(h;g)map:kinetic(

g′ϕ, g′∇) = S

(h;g)map:kinetic(ϕ,∇) .

The other terms in S(ρ,h;Φ,g,B,C)standard (ϕ,∇) do not involve a partially-defined inner product and

hence are all defined. That the integrand inside Tr all transform by conjugation under a gaugesymmetry as for OAzX follows Lemma 4.1.

This proves the theorem.

Remark 4.5. [gauge-fixing condition] As in any gauge field theory (e.g. [P-S]), understanding

how to fix the gauge is an important part of understanding S(ρ,h;Φ,g,B,C)standard (ϕ,∇).

The standard action as an enhanced non-Abelian gauged sigma model

Recall that, in an updated language and in a form for easy comparison, a sigma model (σ-model,SM) on a (Riemannian or Lorentzian) manifold (Y, g) (of dimension n) is a field theory on a(Riemannian or Lorentzian) manifold (X,h) (of some dimension m) with

· Field : differentiable maps f : X → Y ,

· Action functional :

S(h,g)sigma model (f) := ± 1

2

∫X〈df , df〉(g,h) vol h = ±1

2

∫X‖f∗g‖2h vol h

:= ± 12

∫X

m∑µ,ν=1

n∑i,j=1

hµν(x)gij(f(x))∂f i

∂xµ (x)∂f j

∂xν (x)√|deth(x)| dmx ,

38

in terms of local coordinates x = (x1, · · · , xm) on X and y = (y1, · · · , yn) on Y ; cf. [GM-L] andsee e.g. [C-T] for modern update and further references. (The ± sign depends on the signatureof the metric.) At the classical level, this is a theory of harmonic maps; cf. [E-L], [E-S], [Ma],[Sm].

Back to our situation. To begin with, the kinetic term

S(h;g)map:kinetic(ϕ,∇) :=

12 Tm−1

∫X

Re(Tr 〈Dϕ , Dϕ〉(h,g)

)vol h

qualifies the standard action S(ρ,h;Φ,g,B,C)standard (ϕ,∇) to be regarded as a sigma model, now based on

· Field : (∗1)-admissible differentiable maps ϕ : (XAz, E ;∇)→ Y .

The fact that S(ρ,h;Φ,g,B,C)standard (ϕ,∇) is invariant under the gauge symmetry group

Ggauge := C∞(Aut C(E)) and that the latter is non-Abelian justify that this sigma model isindeed a non-Abelian gauged sigma model (nAGSM). However, compared with, for example,the well-studied d = 2, N = (2, 2) (Abelian) gauged linear sigma model, e.g. [H-V] and [Wi1],

the gauge symmetry of S(ρ,h;Φ,g,B,C)standard (ϕ,∇) does not arise from gauging a global group-action

on the target space Y . (For this reason, one may call S(ρ,h;Φ,g,B,C)standard (ϕ,∇) a sigma model with

non-Abelian gauge symmetry as well.) For D-branes, its additional coupling to the backgroundRamond-Ramond field C on Y is essential ([Po1]) and, hence, the Chern-Simons/Wess-Zumino

term S(C,B)CS/WZ (ϕ,∇). Also, we like our dynamical field (ϕ,∇) coupled to the background dilaton

field Φ on Y as well so that the essence of the other important action — the Dirac-Born-Infeldaction — for D-branes can be retained as much as we can. This motivates the dilaton termS

(ρ,h;Φ)dilaton (ϕ). In summary,

S(ρ,h;Φ,g,B,C)standard (ϕ,∇) := S

(ρ,h;Φ,g,B)nAGSM (ϕ,∇) + S

(C,B)CS/WZ (ϕ,∇) + S

(ρ,h;Φ)dilaton (ϕ)

=: S(ρ,h;Φ,g,B,C)

nAGSM+ (ϕ,∇) ,

which explains the name enhanced non-Abelian gauged sigma model (nAGSM+).

5 Admissible family of admissible pairs (ϕT ,∇T )

In this section we introduce the notion of one-parameter admissible families of admissible pairsand rephrase the basic settings and results in Sec. 3.2 in a relative format for such a family. Somecurvature tensor computations are given for later use. The natural generalization (without work)to two-parameter admissible families of admissible pairs is remarked in the last theme of thesection. This prepares us for the study of the variational problem of the enhanced kinetic term

for maps S(ρ,h;Φ,g)

map:kinetic+(ϕ,∇) in the standard action S(ρ,h;Φ,g,B,C)standard (ϕ,∇) for D-branes.

Basic setup and the notion of admissible families of admissible pairs (ϕT ,∇T )

Let T = (−ε, ε) ⊂ R1, with coordinate t and ε > 0 small, be the one-parameter space and∂t := ∂/∂t and dt be respectively the tangent vector field and the 1-form determined by thecoordinate t on T . Let (X,E) be a manifold X of dimension m with a complex vector bundleE of rank r. Recall the structure sheaf OX of X and the OX -module E from E.

Consider the following families of objects over T :

39

· XT := X × T , with the structure sheaf OXT and regarded as the constant family ofmanifolds over T determined by X. XT is equipped with the built-in projection mapsprX : XT → X and prT : XT × T → T . For U ⊂ X an open set, we will denote by UT thecorresponding open set U × T ⊂ X × T over T .

· T∗XT := the tangent bundle of XT and T∗XT := the tangent sheaf of XT ;T ∗XT := the cotangent bundle of XT and T ∗XT := the cotangent sheaf of XT ;T∗(XT /T ) := the relative tangent bundle of XT over T andT∗(XT /T ) := the relative tangent sheaf of XT over T ;T ∗(XT /T ) := the relative cotangent bundle of XT over T andT ∗(XT /T ) := the relative cotangent sheaf of XT over T .When X is endowed with a (Riemannain or Lorentzian) metric h, h induces canonicallyan inner-product structure on fibers of T∗(XT /T ) and its dual, T ∗(XT /T ), over T . Theseinduced inner-product structure will be denoted by 〈 · , ·〉h.

· ET := pr∗XE the pull-back vector bundle of E to XT , regarded as the constant T -family ofvector bundles over X determined by E; and ET := pr∗XE the corresponding OXT -module,regarded as the constant T -family of OX -modules determined by E .The projection map prX : XT → X induces a projection map prE : ET → E between thetotal space of bundles in question. T∗ET (resp. T∗ET ) denotes the tangent space (resp.the tangent sheaf) of the total space of ET .

· (XAzT , ET ) := (XT ,OAzXT := EndO CXT

(ET ), ET ), regarded as the constant T -family of Azu-

maya/matrix manifolds with a fundamental module determined by (XAz, E). There is atrace map

Tr : OAzXT −→ OCXT

as OXT -modules, which takes Id ET to r.

and take the following notational conventions:

· Through the product structure XT = X × T , a vector field ξ (resp. 1-form ω) on X andthe vector field ∂t on T lift canonically to a vector field (resp. 1-form) on XT , which willstill be denoted by ξ (resp. ω) and ∂t respectively.

· For referral, the restriction of XT , XAzT , ET , · · ·T to over t ∈ T will be denoted Xt, X

Azt ,

Et, · · ·t respectively.

Definition 5.1. [connection/covariant derivation trivially flat over T ] A connection∇T on ET (equivalently, connection/covariant derivative ∇T on ET ) is said to be trivially flatover T if the horizontal lifting of ∂t to T∗ET lies in the kernel of the map prE∗ : T∗ET → T∗E.For such a ∇T , we will denote the covariant derivative ∇T∂t simply by ∂t. The curvature tensor

of ∇T will be denoted by F∇T .

Note that any connection on ET is flat over T and hence, due to the topology of T , can bemade trivially flat over T after a bundle-isomorphism. Thus the notion of ‘trivially flat’ is onlya notational convenience for our variational problem, not a true constraint. However, cautionthat while ∇T is always flat over T , its restriction ∇t to Xt varies as t varies in T . Thus, ingeneral, F∇T (∂t, · ) 6= 0.

40

Definition 5.2. [admissible family of admissible pairs (ϕT ,∇T )] A T-family of maps withvarying connections from (XAz, E) to Y is a pair (ϕT ,∇T ), where

ϕT : (XAz, ET ) −→ Y

is a map from (XAzT , ET ) to Y defined contravariantly by a ring-homomorphism

ϕ]T : C∞(Y ) −→ C∞(End C(ET ))

over R ⊂ C and ∇T is a connection on ET that is trivially flat over T . ϕ]T induces a homomor-phism

OY −→ OAzXTbetween equivalence classes of gluing systems of rings, which will still be denoted by ϕ]T .

Let AϕT ⊂ OAzXT = OXT 〈Imϕ]T 〉. Then (ϕT ,∇T ) is said to be a (∗i)-admissible T -family

of (∗j)-admissible pairs if (ϕT ,∇T ) satisfies Admissible Condition (∗i) along T and AdmissibleCondition (∗j) along X, for i, j = 1, 2, 3.

Example 5.3. [(∗2)-admissible T -family of (∗1)-admissible pairs] A (∗2)-admissible T -family of (∗1)-admissible pairs (ϕT ,∇T ) is a T -family of maps ϕT with a varying connection ∇Ttrivially flat over T such that

(∗2) : ∂tComm (AϕT ) ⊂ Comm (AϕT ) and (∗1) : ∇Tξ AϕT ⊂ Comm (AϕT )

for all ξ ∈ T∗(XT /T ). Here, Comm (AϕT ) is the commutant of AϕT in OAzXT .

Three basic OXT -modules with induced structures

Let X be endowed with a (Riemannian or Lorentzian) metric h and Y be endowed with a(Riemannian or Lorentzian) metric g. Denote the canonically induced inner-product structurefrom h and g on whatever bundle applicable by 〈 · , · 〉h and 〈 · , · 〉g respectively. Denote theinduced connection on T∗(XT /T ) and T ∗(XT /T ) by ∇h and the Levi-Civita connection on T∗Yby ∇g. The associated Riemann curvature tensor is denoted by Rh and Rg respectively.

Let (ϕT ,∇T ) be a (∗1)-admissible T -family of (∗1)-admissible pairs. The basic OCXT

-moduleswith induced structures from the setting, as in Sec. 3.2, are listed below to fix notations.

(0) OAzXT : the noncommutative structure sheaf on XT

· The induced connection DT from ∇T , which is also trivially flat over T ,

· An OAzXT -valued, OCX -bilinear (nonsymmetric) inner product from the multiplication

in OAzXT ;

an OCX -valued, OC

X -bilinear (symmetric) inner product after the post-compositionwith Tr .

· Both inner products are covariantly constant with respect to DT and one has theLeizniz rules

DT (m1Tm

2T ) = (DTm1

T )m2T + m1

T DTm2

T ;

dTr (m1Tm

2T ) = TrDT (m1

Tm2T )

= Tr((DTm1

T )m2T

)+ Tr

(m1T D

Tm2T

).

41

(1) T ∗(XT /T )⊗OXTOAzXT

: OAzXT -valued relative 1-forms on XT /T

· The induced connection ∇T,(h,DT ) := ∇h ⊗ Id + Id ⊗DT , trivially flat over T .

· An OAzXT -valued, OCXT

-bilinear (nonsymmetric) inner product 〈 · , · 〉h;

an OCX -valued, OC

X -bilinear (symmetric) inner product Tr 〈 · , · 〉h.

· Both inner products are covariantly constant with respect to ∇T,(h,DT ) and one hasthe Leibniz rules

DT 〈 · , ·′〉h = 〈∇T,(h,DT ) · , ·′〉h + 〈 · , ∇T,(h,DT ) ·′〉g ,dTr 〈 · , ·′〉h = Tr (DT 〈 · , ·′〉h) = Tr 〈∇T,(h,DT ) · , ·′〉h + Tr 〈 · , ∇T,(h,DT ) ·′〉h

for ·, ·′ ∈ T ∗(XT /T )⊗OXTOAzXT

.

(2) ϕ∗TT∗Y := OAzXT ⊗ϕ]T ,OYT∗Y : OAzXT -valued derivations on OY

· The induced connection ∇T,(ϕT ,g) := DT ⊗ Id + Id ·∑n

i=1DTϕ]T (yi)⊗∇g∂

∂yi

(in local

expression), trivially flat over T .

· A partially defined OAzXT -valued, OCX -bilinear (nonsymmetric) inner product 〈 · , · 〉g;

a partially defined OCX -valued, OC

X -bilinear (symmetric) inner product Tr 〈 · , · 〉g.· Both inner products, when defined, are covariantly constant with respect to ∇T,(ϕT ,g)

and one has the Leibniz rules

DT 〈 – , –′〉g = 〈∇T,(ϕT ,g) – , –′〉g + 〈 – , ∇T,(ϕT ,g) –′〉g ,dTr 〈 – , –′〉g = Tr (DT 〈 – , –′〉g) = Tr 〈∇T,(ϕT ,g) – , –′〉g + Tr 〈 – , ∇T,(ϕT ,g) –′〉g ,

whenever all 〈 –′′ , –′′′〉g and Tr 〈 –′′ , –′′′〉g involved are defined.

(3) T ∗(XT /T )⊗OXT ϕ∗TT∗Y : (OAzXT -valued relative 1-form)-valued derivations on OY

This is a combination of the construction in Item (1) and in Item (2).

· The induced connection

∇T,(h,ϕT ,g) = ∇h ⊗ Id ⊗ Id + Id ⊗DT ⊗ Id + Id ⊗ Id ·n∑i=1

DTϕ]T (yi)⊗∇g∂∂yi

(in local expression), trivially flat over T .

· A partially defined OAzXT -valued, OCX -bilinear (nonsymmetric) inner product 〈 · , · 〉(h,g);

a partially defined OCX -valued, OC

X -bilinear (symmetric) inner product Tr 〈 · , · 〉(h,g).· Both inner products, when defined, are covariantly constant with respect to∇T,(h,ϕT ,g)

and one has the Leibniz rules

DT 〈∼ , ∼′〉(h,g) = 〈∇T,(h,ϕT ,g) ∼ , ∼′〉(h,g) + 〈∼ , ∇T,(h,ϕT ,g) ∼′〉(h,g) ,

dTr 〈∼ , ∼′〉(h,g) = Tr (DT 〈∼ , ∼′〉(h,g))

= Tr 〈∇T,(h,ϕT ,g) ∼ , ∼′〉(h,g) + Tr 〈∼ , ∇T,(h,ϕT ,g) ∼′〉(h,g) ,

whenever the 〈∼′′ , ∼′′′〉(h,g) and Tr 〈∼′′ , ∼′′′〉(h,g) involved are defined.

42

Curvature tensors with ∂t and other order-switching formulae

Let (ϕT ,∇T ) be a (∗1)-admissible T -family of (∗1)-admissible pairs. A very basic step in (par-ticularly the second) variational problem involves passing ∂t over a differential operator on Xt’s.In general, a curvature term appears whenever such passing occurs. In this theme, we collectand prove such formulae we need.

First, passing ∂t over a differential operator usually means the appearance of a curvatureterm by the very definition of a curvature tensor:

Lemma 5.4. [curvature tensor with ∂t] Let (ϕT ,∇T ) be a (∗1)-admissible T -family of(∗1)-admissible pairs. Let ξ be a vector field on an open set U ⊂ X small enough so thatϕT (UAzT ) is contained in a coordinate chart on Y , with coordinates (y1, · · · , yn). The standardlifting of ξ to UT is denoted also by ξ. Note that, by construction, [∂t, ξ] = 0 and all ourconnection ∇· in Theme ‘Three basic OAzX -modules with induced structures’ are trivially flat;

hence, F∇·(∂t, ξ) = ∂t∇·ξ − ∇·ξ∂t. One has the following curvature expressions with ∂t on thebasic OXT -modules: (Below we adopt the convention that the Riemann curvature tensor from ametric is denoted by R while the curvature tensor of a connection in all other bundle situationsis denoted by F .)

(01) For sections ωT of T ∗(XT /T ) : R∇h(∂t, ξ)ωT = ∂t∇hξωT − ∇hξ∂tωT = 0.

(02) For sections mT of OAzXT : FDT (∂t, ξ)mT = ∂tDTξ mT − DT

ξ ∂tmT = [(∂t∇T )(ξ),mT ] .

As a consequence of this, if (ϕT ,∇T ) is furthermore a (∗2)-admissible T -family of(∗2)-admissible pairs, then

(∂t∇T )(ξ) ∈ Inn ϕ(∗1)(O

AzX ) i.e. [(∂t∇T )(ξ),AϕT ] ⊂ Comm (AϕT ) .

(1) For sections ωT ⊗mT of T ∗(XT /T )⊗OXT OAzXT

:

F∇T,(h,DT )(∂t, ξ) (ωT ⊗mT )

= ∂t∇T,(h,DT )

ξ (ωT ⊗mT ) − ∇T,(h,DT )

ξ ∂t(ωT ⊗mT ) = ωT ⊗ [(∂t∇T )(ξ),mT ] .

(2) For sections mT ⊗ v of ϕ∗TT∗Y := OAzXT ⊗ϕ]T ,OY T∗Y :

(v on the coordinate chart of Y above, with coordinates (y1, · · · , yn))

F∇T,(ϕT ,g)(∂t, ξ)(mT ⊗ v) = ∂t∇T,(ϕT ,g)ξ (mT ⊗ v) − ∇T,(ϕT ,g)ξ ∂t(mT ⊗ v)

= [(∂t∇T )(ξ),mT ]⊗ v + mT

n∑i=1

[(∂t∇T )(ξ), ϕ]T (yi)]⊗∇g∂∂yi

v

+ mT

n∑i,j=1

(DTξ ϕ

]T (yi) ∂tϕ

]T (yj)⊗∇g∂

∂yj

∇g∂∂yi

v − ∂tϕ]T (yj)DT

ξ ϕ]T (yi)⊗∇g∂

∂yi

∇g∂∂yj

v

).

If (ϕT ,∇T ) is furthermore a (∗2)-admissible T -family of (∗1)-admissible pairs, then thelast term has a Y -coordinate-free form

The last term = mT

∑i,j

∂tϕ]T (yj)Dξϕ

]T (yi)⊗Rg

(∂∂yj

, ∂∂yi

)v = mT

((ϕTR

g)(∂t, ξ))v .

43

(3) For sections ωT ⊗mT ⊗ v of T ∗(XT /T )⊗ ϕ∗TT∗Y := T ∗(XT /T )⊗OXT OAzXT⊗ϕ]T ,OY

T∗Y :

(v on the coordinate chart of Y above, with coordinates (y1, · · · , yn) )

F∇T,(h,ϕT ,g)(∂t, ξ)(ωT ⊗mT ⊗ v)

= ∂t∇T,(h,ϕT ,g)ξ (ωT ⊗mT ⊗ v) − ∇T,(h,ϕT ,g)∂t(ωT ⊗mT ⊗ v)

= ωT ⊗(F∇T,(ϕT ,g)(∂t, ξ)(mT ⊗ v)

).

Proof. Statement (01) follows from the fact that XT is a constant family over T . Statement (02),First Part, follows from a computation with respect to an induced local trivialization of ET froma local trivialization of E

∂tDTξ mT = ∂t

(ξmT + [A∇T (ξ),mT ]

)= ξ∂tmT + [∂tA∇T (ξ),mT ] + [A∇T (ξ), ∂tmT ] = DT

ξ ∂tmT + [(∂t∇T )(ξ),mT ] .

For Second Part, if (ϕT ,∇T ) is furthermore a (∗2)-admissible T-family of (∗2)-admissible pairs,then for f1, f2 ∈ OY , by First Part and the (∗2)-Admissible Condition,[

[(∂t∇T )(ξ), ϕ]T (f1)], ϕ]T (f2)]

= [∂tDTξ ϕ

]T (f1), ϕ]T (f2)] − [DT

ξ ∂tϕ]T (f1), ϕ]T (f2)] = 0 .

Which says that (∂t∇T )(ξ) ∈ Inn ϕT(∗1)(O

AzXT

).

Statement (1) is a consequence of Statement (01) and Statement (02). Statement (3) is aconsequence of Statement (01) and a property of the induced connection on a tensor productof OC

XT-modules with a connection. Let us carry out Statement (2) as a demonstration of the

covariant differential calculus involved.Let mT ⊗ v ∈ ϕ∗TT∗Y . Then, by Statement (02),

∂t∇T,(ϕT ,g)ξ (mT ⊗ v) = ∂t

(DTξ mT ⊗ v + mT

∑i

DTξ ϕ

]T (yi)⊗∇g∂

∂yi

v)

=(DTξ ∂tmT + [(∂t∇T )(ξ),mT ]

)⊗ v + (DT

ξ mT )∑i

∂tϕ]T (yi)⊗∇g∂

∂yi

v

+ (∂tmT )∑i

DTξ ϕ

]T (yi)⊗∇g∂

∂yi

v + mT

∑i

(DTξ ∂tϕ

]T (yi) + [(∂t∇T )(ξ), ϕ]T (yi)]

)⊗∇g∂

∂yi

v

+ mT

∑i,j

DTξ ϕ

]T (yi)∂tϕ

]T (yj)⊗∇g∂

∂yj

∇g∂∂yi

v

while

∇T,(ϕT ,g)ξ ∂t(mT ⊗ v) = ∇T,(ϕT ,g)ξ

(∂tmT ⊗ v + mT

∑i

∂tϕ]T (yi)⊗∇g∂

∂yi

v)

= DTξ ∂tmT ⊗ v + (∂tmT )

∑i

DTξ ϕ

]T (yi)⊗∇g∂

∂yi

v + (DTξ mT )

∑i

∂tϕ]T (yi)⊗∇ ∂

∂yiv

+ mT

∑i

DTξ ∂tϕ

]T (yi)⊗∇g∂

∂yi

v + mT

∑i,j

∂tϕ]T (yi)DT

ξ ϕ]T (yj)⊗∇g∂

∂yj

∇g∂∂yi

v .

Thus,

F∇T,(ϕT ,g)(∂t, ξ)(mT ⊗ v) = (∂t∇T,(ϕT ,g) −∇T,(ϕT ,g)∂t)(mT ⊗ v)

= [(∂t∇T )(ξ),mT ]⊗ v + mT

∑i

[(∂t∇T )(ξ), ϕ]T (yi)]⊗∇g∂∂yi

v

+ mT

∑i,j

(DTξ ϕ

]T (yi)∂tϕ

]T (yj)⊗∇g∂

∂yj

∇g∂∂yi

v − ∂tϕ]T (yj)DT

ξ ϕ]T (yi)⊗∇g∂

∂yi

∇g∂∂yj

v)

44

as claimed, after a relabeling of i, j.

If (ϕT ,∇T ) is furthermore a (∗2)-admissible T -family of (∗1)-admissible pairs, then DTξ ϕ

]T (yi)

and ∂tϕ]T (yj) commute since [DT

ξ ϕ]T (yi), ϕ]T (yj)] = 0 by the (∗1)-Admissible Condition along X

and, hence,

0 = ∂t[DTξ ϕ

]T (yi), ϕ]T (yj)]

= [∂tDTξ ϕ

]T (yi), ϕ]T (yj)] + [DT

ξ ϕ]T (yi), ∂tϕ

]T (yj)] = [DT

ξ ϕ]T (yi), ∂tϕ

]T (yj)]

by the (∗1)-Admissible Condition along X and the (∗2)-Admissible Condition along T . The lastsummand of F∇T,(ϕT ,g)(∂t, ξ)(mT ⊗ v) is then equal to

mT

∑i,j

∂tϕ]T (yj)DT

ξ ϕ]T ⊗

(∇g∂

∂yj

∇g∂∂yi

− ∇g∂∂yi

∇g∂∂yj

)v = mT

((ϕTR

g)(∂t, ξ))v .

This proves the lemma.

The following lemma addresses the issue of passing ∂t over the covariant differential DϕT ofϕT . Though such passing is not a curvature issue in the conventional sense, it does carry a tasteof curvature calculations.

Lemma 5.5. [∂tDTϕT versus ∇T,(ϕT , g)∂tϕT ] Let (ϕT ,∇T ) be a (∗1)-admissible T -family of

(∗1)-admissible pairs. With the above notation and convention, let ξ be a vector field on X.Then, for a chart of Y with coordinates (y1, · · · , yn), one has

∂tDTξ ϕT = ∇T,(ϕT , g)ξ ∂tϕT − (ad⊗∇g)∂tϕTD

Tξ ϕT +

n∑i=1

[(∂t∇T )(ξ), ϕ]T (yi)]⊗ ∂∂yi

.

Here, only as a compact notation,

(ad⊗∇g)∂tϕTDTξ ϕT :=

n∑i,j=1

[∂tϕ](yi), DT

ξ ϕ](yj)]⊗∇g∂

∂yi

∂∂yj

= −n∑

i,j=1

[DTξ ϕ

](yj), ∂tϕ](yi)]⊗∇g∂

∂yj

∂∂yi =: − (ad ⊗∇g)DTξ ϕT ∂tϕT .

If (ϕT ,∇T ) is furthermore a (∗2)-admissible T -family of (∗2)-admissible pairs, then the lastterm has a Y -coordinate-free expression

ad (∂t∇T )(ξ)ϕT .

Proof. Under the given setting and by Lemma 5.4 (02),

∂tDTξ ϕT = ∂t

(∑i

DTξ ϕ

]T (yi)⊗ ∂

∂yi

)=

∑i

(DTξ ∂tϕ

]T (yi) + [(∂t∇T )(ξ), ϕ]T (yi)]

)⊗ ∂

∂yi+∑i,j

DTξ ϕ

]T (yi)∂tϕ

]T (yj)⊗∇g∂

∂yj

∂∂yi

while

∇T,(ϕT ,g)ξ ∂tϕT = ∇T,(ϕT ,g)ξ

(∑i

∂tϕ]T (yi)⊗ ∂

∂yi

)=

∑i

DTξ ∂tϕ

]T (yi)⊗ ∂

∂yi+∑i,j

∂tϕ]T (yi)DT

ξ ϕ]T (yj)⊗∇g∂

∂yj

∂∂yi

.

45

Thus,

∂tDTξ ϕ − ∇

T,(ϕT ,g)ξ ∂tϕT

=∑i

[(∂t∇T )(ξ), ϕ]T (yi)]⊗ ∂∂yi

+∑i,j

DTξ ϕ

]T (yi)∂tϕ

]T (yj)⊗∇g∂

∂yj

∂∂yi−∑i,j

∂tϕ]T (yi)DT

ξ ϕ]T (yj)⊗∇g∂

∂yj

∂∂yi

Either apply the identity ∇g∂∂yj

∂∂yi

= ∇g∂∂yi

∂∂yj

to the second term and relabeling i, j of the third,

or apply the identity ∇g∂∂yj

∂∂yi

= ∇g∂∂yi

∂∂yj

to the third term and relabeling i, j of the second,

∑i,j

DTξ ϕ

]T (yi)∂tϕ

]T (yj)⊗∇g∂

∂yj

∂∂yi−∑i,j

∂tϕ]T (yi)DT

ξ ϕ]T (yj)⊗∇g∂

∂yj

∂∂yi

=∑i,j

[DTξ ϕ

]T (yi) , ∂tϕ

]T (yj)]⊗∇g∂

∂yi

∂∂yj

(:= (ad ⊗∇g)DTξ ϕ∂tϕT

)= −

∑i,j

[∂tϕ]T (yi) , DT

ξ ϕ]T (yj)]⊗∇g∂

∂yi

∂∂yj

(:= − (ad ⊗∇g)∂tϕTDT

ξ ϕT

).

This proves the First Statement in Lemma.The Second Statement in Lemma is a consequence of Corollary 3.1.10 and Lemma 5.4 (02).This proves the lemma.

Before continuing the discussion, we introduce a notion that is needed in the next lemma.

Definition 5.6. [half-torsion tensor Tor12, •∇g ] Recall the torsion tensor Tor∇′ of a connection

∇′ on YTor∇′(v1, v2) := ∇′v1

v2 − ∇′v2v1 − [v1, v2]

for v1, v2 ∈ T∗Y . For the Levi-Civita connection ∇g associated to a metric g on Y , Tor∇g ≡ 0by construction. Thus, in this case, for a Φ ∈ C∞(Y ),

(∇gv1v2 − v1v2)Φ = (∇gv2

v1 − v2v1)Φ

for v1, v2 ∈ T∗Y . This defines a symmetric 2-tensor on Y

Tor12,Φ

∇g : T∗Y ×Y T∗Y −→ OY(v1, v2) 7−→ (∇gv1 v2 − v1v2) Φ

,

called the half-torsion tensor of (the torsion-free connection) ∇g associated to Φ ∈ C∞(Y ).

The following lemma addresses the issue of passing ∂t over ‘evaluation of an OAzXT -valuedderivation on C∞(Y )’, and another similar situation:

Lemma 5.7. [∂t((DTξ ϕT )Φ) versus (∂tD

Tξ ϕT )Φ ; DT

ξ ((∂tϕT )Φ) versus (∇T,(ϕT , g)ξ ∂tϕT )Φ]

Let (ϕT ,∇T ) be a (∗1)-admissible T -family of (∗1)-admissible pairs. Continue the notation and

46

convention in Lemma 5.4. Under the canonical isomorphism OAzXT ⊗ϕ]T ,OY OY ' OAzXT

,

∂t((DT

ξ ϕT )Φ)

=(∂tD

Tξ ϕT

)Φ +

n∑i,j=1

DTξ ϕ

]T (yi) ∂tϕ

]T (yj) ⊗

(∂∂yi

∂∂yj

Φ −(∇g∂

∂yi

∂∂yj

)Φ)

=(∂tD

Tξ ϕT

)Φ −

(ϕTTor

12,Φ

∇g)(ξ, ∂t) ;

and

DTξ

((∂tϕT )Φ

)=

(∇T,(ϕT , g)ξ ∂tϕT

)Φ +

n∑i,j=1

∂tϕ]T (yi)DT

ξ ϕ]T (yj) ⊗

(∂∂yi

∂∂yj

Φ −(∇g∂

∂yi

∂∂yj

)Φ)

=(∇T,(ϕT , g)ξ ∂tϕT

)Φ −

(ϕTTor

12,Φ

∇g)(∂t, ξ) .

Proof. For the first identity,

∂t((DT

ξ ϕT )Φ)

= ∂t

(∑i

DTξ ϕ

]T (yi)⊗ ∂

∂yiΦ)

=∑i

∂tDTξ ϕ

]T ⊗

∂∂yi

Φ +∑i,j

DTξ ϕ

]T (yi)∂tϕ

]T (yj)⊗ ∂

∂yj∂∂yi

Φ

while (∂tD

Tξ ϕT

)Φ =

(∂t∑i

DTξ ϕ

]T (yi)⊗ ∂

∂yi

)Φ

=(∑

i

∂tDTξ ϕ

]T (yi)⊗ ∂

∂yi+∑i,j

DTξ ϕ

]T (yi)∂tϕ

]T (yj)⊗∇g∂

∂yj

∂∂yi

)Φ .

Thus,

∂t((DT

ξ ϕT )Φ)−(∂tD

Tξ ϕT

)Φ

=∑i,j

DTξ ϕ

]T (yi)∂tϕ

]T (yj)⊗

(∂∂yj

∂∂yi− ∇g∂

∂yj

∂∂yi

)Φ

=∑i,j

DTξ ϕ

]T (yi)∂tϕ

]T (yj)⊗

(∂∂yi

∂∂yj− ∇g∂

∂yi

∂∂yj

)Φ = −

(ϕTTor

12 ,Φ

∇g)(ξ, ∂t)

and the first identity follows.For the second identity,

DTξ

((∂tϕT )Φ

)= DT

ξ

(∑i

∂tϕ]T (yi)⊗ ∂

∂yiΦ)

=∑i

DTξ ∂tϕ

]T (yi)⊗ ∂

∂yiΦ +

∑i,j

∂tϕ]T (yi)DT

ξ ϕ]T (yj)⊗ ∂

∂yj∂∂yi

Φ

while (∇T,(ϕT , g)ξ ∂tϕT

)Φ =

(∇T,(ϕT ,g)ξ

∑i

∂tϕ]T (yi)⊗ ∂

∂yi

)Φ

=(∑

i

DTξ ∂tϕ

]T (yi)⊗ ∂

∂yi+∑i,j

∂tϕ]T (yi)DT

ξ ϕ]T (yj)⊗∇g∂

∂yj

∂∂yi

)Φ .

47

Thus,

DTξ

((∂tϕT )Φ

)−(∇T,(ϕT , g)ξ ∂tϕT

)Φ

=∑i,j

∂tϕ]T (yi)DT

ξ ϕ]T (yj)⊗

(∂∂yj

∂∂yi− ∇g∂

∂yj

∂∂yi

)Φ

=∑i,j

∂tϕ]T (yi)DT

ξ ϕ]T (yj)⊗

(∂∂yi

∂∂yj− ∇g∂

∂yi

∂∂yj

)Φ = −

(ϕTTor

12 ,Φ

∇g)(∂t, ξ)

and the second identity follows.This proves the lemma.

Remark 5.8. [for (∗2)-admissible family of (∗1)-admissible pairs ] If (ϕT ,∇T ) is furthermorea (∗2)-admissible T -family of (∗1)-admissible pairs, then, as in the proof of Lemma 5.4 (2),

DTξ ϕ

]T (yi) and ∂tϕ

]T (yj) commute for all i, j. In this case,

(ϕTor

12,Φ

∇g)(ξ, ∂t) =

(ϕTor

12,Φ

∇g)(∂t, ξ) .

Two-parameter admissible families of admissible pairs

Let T = (−ε, ε)2 ⊂ R2, ε > 0 small, be a two-parameter space with coordinates (s, t). Thesetting and results above for one-parameter admissible families of admissible pairs generalizeswithout work to two-parameter admissible of admissible pairs. In particular,

Definition 5.9. [two-parameter admissible family of admissible pairs] A (∗2)-admissibleT -family of (∗1)-admissible maps is a (∗1)-admissible map ϕT : (XAzT , ET ;∇T )→ Y , where ET istrivially flat over T , such that ∂sCommAϕT ⊂ Comm (AϕT ) and ∂tCommAϕT ⊂ Comm (AϕT ).

The following is a consequence of the proof of Lemma 3.2.2.5:

Lemma 5.10. [symmetry property of Tr 〈F∇T,(ϕT ,g)(∂s, ξ2)∂tϕT , DTξ4ϕT 〉] Let

ϕT : (XAzT , ET ;∇T )→ Y be a (∗2)-admissible T -family of (∗1)-admissible maps. Let ξ2, ξ4 ∈ T∗Xand denote the same for their respective lifting to T∗(XT /T ). Then,

Tr 〈F∇T,(ϕT ,g)(∂s, ξ2)∂tϕT , DTξ4ϕT 〉g = −Tr 〈∂tϕT , F∇T,(ϕT ,g)(∂s, ξ2)Dξ4ϕT 〉g

= −Tr 〈F∇T,(ϕT ,g)(∂s, ξ2)DTξ4ϕT , ∂tϕT 〉g = Tr 〈F∇T,(ϕT ,g)(ξ2, ∂s)D

Tξ4ϕT , ∂tϕT 〉g .

Proof. Let ξ be ξ2 or ξ4. Since ∂sComm (AϕT ) ⊂ Comm (AϕT ) and and ∂ξAϕT ⊂ Comm (AϕT ),both ∂sD

Tξ ϕT and ∂s∂tϕT lie in Comm (AϕT )⊗ϕ],OYT∗Y . Locally explicitly,

∂sDTξ ϕT =

∑i

∂sDTξ ϕ

]T (yi)⊗ ∂

∂yi +∑i,j

DTξ ϕ

]T (yi) ∂sϕ

]T (yj)⊗∇g∂

∂yj

∂∂yi ;

∂s∂tϕT =∑i

∂s∂tϕ]T (yi)⊗ ∂

∂yi +∑i,j

∂tϕ]T (yi) ∂sϕ

]T (yj)⊗∇g∂

∂yj

∂∂yi .

48

Now follow the proof of Lemma 3.2.2.5, but under only the (∗1)-Admissible Condition on(ϕT ,∇T ), to convert Tr 〈F∇T,(ϕT ,g)(∂s, ξ2)∂tϕT , D

Tξ4ϕT 〉g to Tr 〈∂tϕT , F∇T,(ϕT ,g)(∂s, ξ2)Dξ4ϕT 〉g .

Since Tr 〈 – , –′ 〉g is defined as long as one of –, –′ is in Comm (AϕT ) ⊗ϕ]T ,OY

T∗Y , one realizes

that all the terms that appear in the process via the Leibniz rule are defined except

−Tr 〈∇T,(ϕT ,g)ξ2∂tϕT , ∂sD

Tξ4ϕT 〉g + Tr 〈∂s∂tϕT , ∇T,(ϕT ,g)ξ2

DTξ4ϕT 〉g .

Under the additional (∗2)-Admissible Condition on (ϕT ,∇T ) along T , both ∂sDTξ4ϕT and ∂s∂tϕT

now lie in Comm (AϕT )⊗ϕ],OYT∗Y ; and the above two exceptional terms become defined.The lemma follows.

6 The first variation of the enhanced kinetic term for maps and......

Let (ϕ,∇) be a (∗1)-admissible pair. Recall the setup in Sec. 5. Let T = (−ε, ε) ⊂ R1, for someε > 0 small, and (ϕT ,∇T ) be a (∗1)-admissible T -family of (∗1)-admissible pairs that deforms(ϕ,∇) = (ϕT ,∇T )|t=0. We derive in Sec. 6.1 and Sec. 6.2 the first variation formula of the newlyintroduced enhanced kinetic term for maps

S(ρ,h;Φ,g)

map:kinetic+(ϕ,∇) :=1

2Tm−1

∫X

Re Tr 〈Dϕ , Dϕ〉(h,g) vol h +

∫X

Re Tr 〈dρ, ϕdΦ〉h vol h

in the standard action for D-branes. As the ‘taking the real part’ operation Re (· · · · · · ) is aOX -linear operation and can always be added back in the end, we will consider

S(ρ,h;Φ,g)

map:kinetic+(ϕ,∇)C :=1

2Tm−1

∫X

Tr 〈Dϕ , Dϕ〉(h,g) vol h +

∫X

Tr 〈dρ, ϕdΦ〉h vol h

so that we don’t have to carry Re around.The first variation of the gauge/Yang-Mills term is analogous to that in the ordinary Yang-

Mills theory and the first variation of the Chern-Simons/Wess-Zumino term is an update from[L-Y8: Sec. 6] (D(13.1)). Both are given in Sec. 6.3 under the stronger (∗2)-Admissible Condition.

6.1 The first variation of the kinetic term for maps

Recall the (complexified) kinetic energy E∇t(ϕt)

C of ϕt for a given ∇t, t ∈ T := (−ε, ε),

E∇t(ϕt)

C := S(h;g)map:kinetic(ϕt,∇

t)C :=1

2Tm−1

∫X

Tr 〈Dtϕt , Dtϕt〉(h,g) vol h .

As t varies, with a slight abuse of notation, denote the resulting function of t by

E∇T

(ϕT )C := S(h;g)map:kinetic(ϕT ,∇

T )C :=1

2Tm−1

∫X

Tr 〈DTϕT , DTϕT 〉(h,g) vol h ,

with the understanding that all expressions are taken on Xt with t varying in T .

49

Let U ⊂ X be an open set with an orthonormal frame (eµ)µ=1, ··· ,m. Let (eµ)µ=1, ··· ,m be thedual co-frame. Assume that U is small enough so that ϕT (UAzT ) is contained in a coordinatechart of Y , with coordinates (y1, · · · , yn). Then, over U ,

d

dtE∇

T

(ϕT )C =1

2Tm−1

∫U

∂tTr 〈DTϕT , DTϕT 〉(h,g) vol h

=1

2Tm−1

∫U

Tr ∂t〈DTϕT , DTϕT 〉(h,g)vol h

=1

2Tm−1

∫U

Tr ∂t

m∑µ=1

〈DTeµϕT , D

TeµϕT 〉gvol h

= Tm−1

∫U

Tr

m∑µ=1

〈∂tDTeµϕT , D

TeµϕT 〉g vol h

= Tm−1

∫U

Tr∑µ

〈∇T,(ϕT ,g)eµ ∂tϕT , DTeµϕT 〉g vol h

+ Tm−1

∫U

Tr∑µ

〈(ad ⊗∇g)DTeµϕT ∂tϕT , DTeµϕT 〉g vol h

+ Tm−1

∫U

∑µ

〈n∑i=1

[(∂t∇T )(eµ), ϕ]T (yi)]⊗ ∂∂yi , D

TeµϕT 〉g vol h

= (I.1) + (I.2) + (I.3) .

(I.1) = Tm−1

∫U

∑µ

Tr(DTeµ〈∂tϕT , D

TeµϕT 〉g − 〈∂tϕT , ∇

T,(ϕT ,g)eµ DT

eµϕT 〉g)

vol h

= Tm−1

∫U

∑µ

eµTr 〈∂tϕT , DTeµϕT 〉g vol h + Tm−1

∫U

Tr 〈∂tϕT , −∑µ∇

T,(ϕT ,g)eµ DT

eµϕT 〉g vol h

= (I.1.1) + (I.1.2) .

Summand (I.1.1) suggests a boundary term. To really extract the boundary term from it,consider the T -family of C-valued 1-forms on U

αT(I, ∂tϕT ) := Tr 〈∂tϕT , DTϕT 〉g ,

which depends C∞(U)C-linearly on ∂tϕT . Let

ξT(I, ∂tϕT ) :=

m∑µ=1

(Tr 〈∂tϕT , DT

eµϕT 〉g)eµ

be the T -family of dual C-valued vector fields of αT(I, ∂tϕT ) on U with respect to the metric h.

Note that ξT(I, ∂tϕT ) depends C∞(U)C-linearly on ∂tϕT as well. Then

(I.1.1) = Tm−1

∫U

∑µ

eµ〈ξT(I, ∂tϕT ) , eµ〉h vol h

= Tm−1

∫U

∑µ

〈∇heµξT(I, ∂tϕT ) , eµ〉h vol h + Tm−1

∫U〈ξT(I, ∂tϕT ) ,

∑µ∇

heµeµ〉h vol h .

The first term is equal to

Tm−1

∫U

(− div ξT(I, ∂tϕT )) vol h = Tm−1

∫Ud iξT

(I, ∂tϕT )vol h = Tm−1

∫∂UiξT

(I, ∂tϕT )vol h ,

50

which is the sought-for boundary term, whose integrand satisfies the requirement that it beC∞(U)C-linear on ∂tϕT . The second term is equal to

Tm−1

∫U

Tr 〈∂tϕT , DT∑µ∇heµeµ

ϕT 〉g vol h

by construction, which is C∞(U)C-linear in ∂tϕT and hence in a final form.The integrand of Summand (I.1.2) is already C∞(U)C-linear in ∂tϕT and hence in a final

form.Summand (I.2) can be re-written as

(I.2) = −Tm−1

∫U

Tr∑µ

〈(ad ⊗∇g)∂tϕTDTeµϕT , D

TeµϕT 〉g vol h .

Thus, its integrand is already C∞(U)C-linear in ∂tϕT and hence in a final form.Finally, since the built-in inclusion OC

U ⊂ OAzU identifies OCU with the center of OAzU , Summand

(I.3) is C∞(U)C-linear and hence in its final fom.

Altogether, we almost complete the calculation except the issue of whether all the innerproducts Tr 〈 · , · 〉g that appear in the procedure are truly defined. For this, one notices thatwherever such an inner product appears above, at least one of its arguments is either ∂tϕT orDTeµϕT , for some µ. It follows from Lemma 3.2.2.4 that they are indeed defined.In summary,

Proposition 6.1.1. [first variation of kinetic term for maps] Let (ϕT ,∇T ) be a (∗1)-admissible T -family of (∗1)-admissible pairs. Then,

d

dtE∇

T(ϕT )C =

d

dt

(1

2Tm−1

∫U

Tr 〈DTϕT , DTϕT 〉(h,g) vol h

)= Tm−1

∫∂UiξT

(I, ∂tϕT )vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

(DT∑m

µ=1∇heµeµ−∑m

µ=1∇T,(ϕT ,g)eµ DT

eµ

)ϕT⟩g

vol h

− Tm−1

∫U

Trm∑µ=1

〈(ad ⊗∇g)∂tϕTDTeµϕT , D

TeµϕT 〉g vol h

+ Tm−1

∫U

m∑µ=1

〈n∑i=1

[(∂t∇T )(eµ), ϕ]T (yi)]⊗ ∂∂yi

, DTeµϕT 〉g vol h .

Here, the first summand is the boundary term with ξT(I, ∂tϕT ) :=∑m

µ=1(Tr 〈∂tϕT , DTeµϕT 〉g)eµ

C∞(U)C-linear in ∂tϕT ; the integrand of the second and the third terms are C∞(U)C-linear in∂tϕT and their real part contribute first-order and second-order terms to the equations of motionfor (ϕ,∇); the integrand of the last term is C∞(U)C-linear in ∂t∇T and its real part contributesterms, first order in ϕ but zeroth order in the connection 1-from of ∇, to the equations of motion

for (ϕ,∇) in addition to those from the first variation of the rest part of S(ρ,h;Φ,g,B,C)standard (ϕ,∇).

These lower-order terms contribute to the equations of motion for (ϕ,∇) but do not change thesignature of the system.

51

Remark 6.1.2. [for (∗2)-admissible T -family of (∗2)-admissible pairs] If furtheremore (ϕ,∇) is(∗2)-admissible and (ϕT ,∇T ) is a (∗2)-admissible T -family of (∗2)-admissible pairs that deforms(ϕ,∇), then the third summand of the first variation formula in Proposition 6.1.1 vanishes andthe fourth/last summand has a Y -coordinate-free form

Tm−1

∫U

m∑µ=1

〈ad (∂t∇T )(eµ)ϕT , DTeµϕT 〉g vol h .

In this case, the first variation with respect to ϕ alone (i.e. setting ∂t∇T = 0), cf. the first twosummands, takes the form of a direct formal generalization of the first variation formula in thestudy of harmonic maps; e.g. [E-L], [E-S], [Ma], [Sm].

6.2 The first variation of the dilaton term

We now turn to the (complexified) dilaton term in S(ρ,h;Φ,g,B,C)standard (ϕ,∇)C.

Let ϕT : (XAz, ET ;∇T ) → Y be a (∗1)-admissible T -family of (∗1)-admissible pairs. Then,over an open set U ⊂ X,

S(ρ,h;Φ)dilaton (ϕT )C =

∫U

Tr 〈dρ , ϕTdΦ〉h vol h

=

∫U

Tr

m∑µ=1

(dρ(eµ)

n∑i=1

DTeµϕ

]T (yi)ϕ]T

( ∂Φ

∂yi

))vol h

=

∫U

Tr

(∑µ

dρ(eµ) ((DTeµϕT )Φ)

)vol h .

d

dtS

(ρ,h;Φ)

dilaton(ϕT )C =

∫U

Tr

m∑µ=1

dρ(eµ) ∂t

((DT

eµϕT )Φ)

vol h

=

∫U

Tr∑µ

dρ(eµ)(

(∂tDTeµϕT )Φ

)vol h

+

∫U

Tr∑µ

dρ(eµ)

n∑i,j=1

DTeµϕ

]T (yi)∂tϕ

]T (yj)⊗

(∂∂yj

∂∂yiΦ−

(∇g∂

∂yj

∂∂yi

)Φ

)vol h

= (II.1) + (II.2) .

The integrand of Summand (II.2) is C∞(U)C-linear in ∂tϕT and hence in a final form.

(II.1) =

∫U

Tr∑µ

dρ(eµ)((∇T,(ϕT , g)eµ ∂tϕT

)Φ)

vol h

−∫U

Tr∑µ

dρ(eµ)((

(ad ⊗∇g)∂tϕTDTeµϕT

)Φ)

vol h

+

∫U

Tr∑µ

dρ(eµ)((∑

i

[(∂t∇T )(eµ) , ϕ]T (yi)

]⊗ ∂

∂yi

)Φ)

vol h

= (II.1.1) + (II.1.2) + (II.1.3) .

Both Summand (II.1.2) and Summand (II.1.3) vanish since

Tr ([a, b]c) = 0 if [b, c] = 0

52

for r × r matrices a, b, c.

(II.1.1) =

∫U

Tr∑µ

dρ(eµ)DTeµ((∂tϕT )Φ)) vol h

−∫U

Tr∑µ

dρ(eµ)

n∑i,j=1

∂tϕ]T (yi)DT

eµϕ]T (yj) ⊗

(∂∂yi

∂∂yj Φ −

(∇g∂

∂yi

∂∂yj

)Φ

)vol h

= (II.1.1.1) + (II.1.1.2) .

The integrand of Summand (II.1.1.2) is C∞(U)C-linear in ∂tϕT and hence in a final form. Itcan be combined with Summand (II.2) to give

(II.1.1.2) + (II.2)

= −∫U

Tr∑µ

dρ(eµ)

n∑i,j=1

[∂tϕ]T (yi) , DT

eµϕ]T (yj)]⊗

(∂∂yi

∂∂yj Φ −

(∇g∂

∂yi

∂∂yj

)Φ

)vol h ,

which again vanishes due to Tr .

(II.1.1.1) =

∫U

∑µ

dρ(eµ)TrDTeµ((∂tϕT )Φ) vol h

=

∫U

∑µ

dρ(eµ) eµTr ((∂tϕT )Φ) vol h

=

∫U

∑µ

eµ

(dρ(eµ) Tr ((∂tϕT )Φ)

)vol h −

∫U

(∑µ

eµdρ(eµ))

Tr ((∂tϕT )Φ) vol h

= (II.1.1.1.1) + (II.1.1.1.2)

The integrand of Summand (II.1.1.1.2) is C∞(U)C-linear in ∂tϕT and hence in a final form.To extract the boundary term from Summand (II.1.1.1.1), consider the T -family of C-valued1-forms on U

αT(II,∂tϕT ) := dρTr ((∂tϕT )Φ) ,

which depends C∞(U)C-linearly on ∂tϕT . Let

ξT(II,∂tϕT ) =m∑µ=1

(dρ(eµ) Tr ((∂tϕT )Φ)

)eµ

be the T-family of dual C-valued vector fields of αT(II,∂tϕT ) on U with respect to the metric h.

Note that ξT(II,∂tϕT ) depends C∞(U)C-linearly on ∂tϕT as well. Then

(II.1.1.1.1) =

∫U

∑µ

eµ〈ξT(II,∂tϕT ) , eµ〉h vol h

=

∫U

∑µ

〈∇heµξT(II,∂tϕT ) , eµ〉h vol h +

∫U

〈ξT(II,∂tϕT ) ,∑µ∇

heµeµ〉h vol h

The first term is equal to∫U

(− div ξT(II, ∂tϕT )) vol h =

∫Ud iξT

(II, ∂tϕT )vol h =

∫∂UiξT

(II, ∂tϕT )vol h ,

53

which is the sought-for boundary term, whose integrand satisfies the requirement that it beC∞(U)C-linear in ∂tϕT . The second term is equal to∫

Udρ(∑

µ∇heµeµ)

Tr ((∂tϕT )Φ) vol h

by construction, which is C∞(U)C-linear in ∂tϕT and hence in a final form.In summary,

Proposition 6.2.1. [first variation of dilaton term] Let (ϕT ,∇T ) be a (∗1)-admissibleT -family of (∗1)-admissible pairs. Then,

d

dtS

(ρ,h;Φ)

dilaton(ϕT )C =d

dt

∫U

Tr 〈dρ , ϕT dΦ〉h vol h

=

∫∂U

iξT(II, ∂tϕT )

vol h

+

∫U

(dρ(∑m

µ=1∇heµeµ)−∑mµ=1eµdρ(eµ)

)Tr ((∂tϕT )Φ) vol h .

Here, the first summand is the boundary term with ξT(II,∂tϕT ) :=∑m

µ=1(dρ(eµ) Tr ((∂tϕT )Φ))eµ

C∞(U)C-linear in ∂tϕT ; the integrand of the second summand C∞(U)C-linear in ∂tϕT and theycontribute additional zeroth-order terms to the equations of motion for (ϕ,∇). In particular,while the dilaton term of the standard action modifies the equations of motion for (ϕ,∇), it doesnot change the signature of the system.

6.3 The first variation of the gauge/Yang-Mills term and the Chern-Simons/Wess-Zumino term

To make sure that differential forms on Y of rank ≥ 2 are pull-pushed to (OAzX -valued-)differentialforms on X (cf. Lemma 2.1.11), we assume in this subsection that ϕT : (XAzT , ET ;∇T )→ Y is a(∗2)-family of (∗2)-admissible maps. (Note that as the gauge/Yang-Mills term is defined througha norm-squared, (∗1)-admissible family of (∗1)-admissible (ϕT ,∇T ) is enough for the derivationof the first variation formula of the gauge/Yang-Mills term but the result will be slightly messier.)

6.3.1 The first variation of the gauge/Yang-Mills term

Let (e1, · · · , em) be an orthonormal frame on U . Then, over U ,

S(h;B)gauge /YM(ϕT ,∇T )C := − 1

2

∫U

Tr ‖2πα′F∇T + ϕTB‖2h vol h

= − 12

∫U

Tr∑µ,ν

((2πα′F∇T + ϕTB)(eµ, eν)

)2vol h .

Applying the following basic identities:

∂tF∇T (eµ, eν) = DTeµ

((∂t∇T )(eν)

)− DT

eν

((∂t∇T )(eµ)

)− (∂t∇T )([eµ, eν ]) ,

∂t((ϕTB)(eµ, eν)

)=

∑i,j

∂t(ϕ]T (Bij)

)DTeµϕ

]T (yi)DT

eνϕ]T (yj)

+∑i,j

ϕ]T (Bij)(DTeµ∂tϕ

]T (yi) +

[(∂t∇T )(eµ), ϕ]T (yi)

])DTeνϕ

]T (yj)

+∑i,j

ϕ]T (Bij)DTeµϕ

]T (yi)

(DTeν∂tϕ

]T (yj) +

[(∂t∇T )(eν), ϕ]T (yj)

]).

54

and proceeding similarly to Sec. 6.1, one has the following results.

d

dtS

(h;B)gauge /YM

(ϕT ,∇T )C = − 12

∫U

Tr ∂t∑µ,ν

((2πα′F∇T + ϕTB)(eµ, eν)

)2

vol h

= −∫U

Tr∑µ,ν

∂t

((2πα′F∇T + ϕTB)(eµ, eν)

)·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

= −∫U

Tr∑µ,ν

2πα′∂t(F∇T (eµ, eν)

)·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

−∫U

Tr∑µ,ν

∂t(ϕTB(eµ, eν)

)·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

= (III.1) + (III.2) .

(III.1) := −∫U

Tr∑µ,ν

2πα′∂t(F∇T (eµ, eν)

)·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

= − 2πα′∫U

Tr∑µ,ν

(DTeµ

((∂t∇T )(eν)

)− DT

eν

((∂t∇T )(eµ)

)− (∂t∇T )([eµ, eν ])

)·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

= − 4πα′∫∂U

iξT(III,∂t∇T )

vol h

− 4πα′∫U

Tr∑ν

(∂t∇T )(eν) ·(

(2πα′F∇T + ϕTB)(∑µ∇

heµeµ, eν)

−∑µ

DTeµ

((2πα′F∇T + ϕTB)(eµ, eν)

)− 1

2

∑µ,λ

eν([eµ, eλ])(2πα′F∇T + ϕTB)(eµ, eλ))

vol h .

Here,

ξT(III,∂t∇T ) :=∑µ,ν

Tr(

(∂t∇T )(eν) · (2πα′F∇T + ϕTB)(eµ, eν))eµ ∈ T∗(UT /T )C

is OCU -linear in ∂t∇T ; and the second summand contributes to the equations of motion for (ϕ,∇).

The latter are standard terms from non-Abelian Yang-Mills theory with additional terms fromϕB.

(III.2) := −∫U

Tr∑µ,ν

∂t(ϕTB(eµ, eν)

)·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

= −∫U

Tr∑µ,ν

(∑i,j

∂t(ϕ]T (Bij)

)DTeµϕ

]T (yi)DT

eνϕ]T (yj)

+∑i,j

ϕ]T (Bij)(DTeµ∂tϕ

]T (yi) +

[(∂t∇T )(eµ), ϕ]T (yi)

])DTeνϕ

]T (yj)

+∑i,j

ϕ]T (Bij)DTeµϕ

]T (yi)

(DTeν∂tϕ

]T (yj) +

[(∂t∇T )(eν), ϕ]T (yj)

]))·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

= (III.2.1) + +((III.2.2.1) + (III.2.2.2)

)+ (III.2.3.1) + (III.2.3.2)

in the order of the appearance of the five summands after the expansion.

55

(III.2.1) := −∫U

Tr∑µ,ν

∑i,j

∂t(ϕ]T (Bij)

)DTeµϕ

]T (yi)DT

eνϕ]T (yj)

·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

:= −∫U

Tr∑µ,ν

∑i,j

((∂tϕT )Bij

)DTeµϕ

]T (yi)DT

eνϕ]T (yj)

·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

has an integrand OCU -linear in ∂tϕT and hence in a final form.

(III.2.2.1) + (III.2.3.1)

:= −∫U

Tr∑µ,ν

(∑i,j

ϕ]T (Bij)DTeµ∂tϕ

]T (yi)DT

eνϕ]T (yj)

+∑i,j

ϕ]T (Bij)DTeµϕ

]T (yi)DT

eν∂tϕ]T (yj)

)·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

= − 2

∫U

Tr∑µ,ν

∑i,j

DTeµ∂tϕ

]T (yi)ϕ]T (Bij)D

Teνϕ

]T (yj)

·((2πα′F∇T + ϕTB)(eµ, eν)

)vol h

= − 2

∫∂U

iξT(III,∂tϕT )

vol h

− 2

∫U

Tr∑ν

∑i,j

∂tϕ]T (yi)(

ϕ]T (Bij)DTeνϕ

]T (yj) ·

((2πα′F∇T + ϕTB)(

∑µ∇

heµeµ, eν)

)−∑µ

DTeµ

(ϕ]T (Bij)D

Teνϕ

]T (yj) ·

((2πα′F∇T + ϕTB)(eµ, eν)

)))vol h .

Here,

ξT(III,∂tϕT ) :=∑µ

(∑ν

∑i,j

∂tϕ]T (yi)ϕ]T (Bij)D

Teνϕ

]T (yj) ·

((2πα′F∇T + ϕTB)(eµ, eν)

))eµ

in T∗(UT /T )C is OCU -linear in ∂tϕT ; and the second summand contributes to

δS(ρ,h;Φ,g,B,C)standard (ϕ,∇)/δϕ-part of the equations of motion for (ϕ,∇).

(III.2.2.2) + (III.2.3.2)

:= −∫U

Tr∑µ,ν

(∑i,j

ϕ]T (Bij)[(∂t∇T )(eµ), ϕ]T (yi)

]DTeνϕ

]T (yj)

+∑i,j

ϕ]T (Bij)DTeµϕ

]T (yi)

[(∂t∇T )(eν), ϕ]T (yj)

])·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h .

has an integrand OCU -linear in ∂t∇T and hence in a final form.

In summary,

56

Proposition 6.3.1.1. [first variation of gauge/Yang-Mills term] Let (ϕT ,∇T ) be a (∗2)-admissible family of (∗2)-admissible pairs. Then

ddt S

(h;B)gauge /YM

(ϕT ,∇T )C = − 12ddt

∫U

Tr ‖2πα′F∇T + ϕTB‖2h vol h

= − 4πα′∫∂U

iξT(III,∂t∇T )

vol h − 2

∫∂U

iξT(III,∂tϕT )

vol h

− 4πα′∫U

Tr∑ν

(∂t∇T )(eν) ·(

(2πα′F∇T + ϕTB)(∑µ∇

heµeµ, eν)

−∑µ

DTeµ

((2πα′F∇T + ϕTB)(eµ, eν)

)− 1

2

∑µ,λ

eν([eµ, eλ])(2πα′F∇T + ϕTB)(eµ, eλ))

vol h .

−∫U

Tr∑µ,ν

(∑i,j

ϕ]T (Bij)[(∂t∇T )(eµ), ϕ]T (yi)

]DTeνϕ

]T (yj)

+∑i,j

ϕ]T (Bij)DTeµϕ

]T (yi)

[(∂t∇T )(eν), ϕ]T (yj)

])·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

−∫U

Tr∑µ,ν

∑i,j

((∂tϕT )Bij

)DTeµϕ

]T (yi)DT

eνϕ]T (yj)

·(

(2πα′F∇T + ϕTB)(eµ, eν))

vol h

− 2

∫U

Tr∑ν

∑i,j

∂tϕ]T (yi)(

ϕ]T (Bij)DTeνϕ

]T (yj) ·

((2πα′F∇T + ϕTB)(

∑µ∇

heµeµ, eν)

)−∑µ

DTeµ

(ϕ]T (Bij)D

Teνϕ

]T (yj) ·

((2πα′F∇T + ϕTB)(eµ, eν)

)))vol h .

Here,

ξT(III,∂t∇T ) :=∑µ,ν

Tr(

(∂t∇T )(eν) · (2πα′F∇T + ϕTB)(eµ, eν))eµ ,

ξT(III,∂tϕT ) :=∑µ

(∑ν

∑i,j

∂tϕ]T (yi)ϕ]T (Bij)D

Teνϕ

]T (yj) ·

((2πα′F∇T + ϕTB)(eµ, eν)

))eµ

in T∗(UT /T )C, with the first OCU -linear in ∂t∇T and the second OC

U -linear in ∂tϕT .

6.3.2 The first variation of the Chern-Simons/Wess-Zumino term for lower dimen-sional D-branes

This is an update of [L-Y8: Sec.6.2] (D(13.1)) in the current setting. Let ϕT : (XAz, ET ;∇T )→Y be an (∗2)-family of (∗2)-admissible maps. We work out the first variation of the Chern-

Simons/Wess-Zumino term S(C,B)CS/WZ (ϕ,∇) for the cases where m := dimX = 0, 1, 2, 3. As the

details involve no identities or techniques that have not yet been used in Sec. 6.1, Sec. 6.2, and/orSec. 6.3.1, we only summarize the final results below.

57

6.3.2.1 D(−1)-brane world-point (m = 0)

For a D(−1)-brane world-point, dimX = 0, ∇ = 0, and S(C(0))

CS/WZ (ϕT ) = T−1 · Tr (ϕ]T (C(0))). It

follows thatddtS

(C(0))CS/WZ (ϕT ) = T−1 Tr ∂t(ϕ

]T (C(0))) = T−1 Tr

((∂tϕT )C(0)

).

6.3.2.2 D-particle world-line (m = 1)

For a D-particle world-line, dimX = 1. Let e1 be the orthonormal frame on an open set U ⊂ X;e1 its dual co-frame. Then, over U ,

S(C(1))CS/WZ (ϕT )C = T0

∫U

TrϕTC(1) = T0

∫U

Tr( n∑i=1

ϕ]T (Ci) ·DTe1ϕ

]T (yi)

)e1 .

It follows that

ddtS

(C(1))CS/WZ (ϕ) = T0

(Tr∑i

∂tϕ]T (yi)ϕ]T (Ci)

)∣∣∂U

− T0

∫U

Tr(∑

i

∂tϕ]T (yi)DT

e1ϕ]T (Ci)

)e1 + T0

∫U

Tr(∑

i

De1ϕ]T (yi) · (∂tϕT )Ci

)e1 .

6.3.2.3 D-string world-sheet (m = 2)

Denote

C(2) := C(2) + C(0)B =∑ij

(Cij + C(0)Bij) dyi ⊗ dyj =

∑i,j

Cijdyi ⊗ dyj

in a local coordinate (y1, · · · , yn) of Y . For a D-string world-sheet, dimX = 2. Let (e1, e2) bean orthonormal frame on an open set U ⊂ X; (e1, e2) its dual co-frame. Then, over U ,

S(C(0),C(2),B)CS/WZ (ϕT ,∇T )C

= T1

∫U

Tr( n∑i,j=1

ϕ]T (Cij)DTe1ϕ

]T (yi)DT

e2ϕ]T (yj)

+πα′ϕ]T (C(0))F∇T (e1, e2) + πα′F∇T (e1, e2)ϕ]T (C(0)))e1 ∧ e2

= T1

∫U

Tr( n∑i,j=1

ϕ]T (Cij)DTe1ϕ

]T (yi)DT

e2ϕ]T (yj) + 2πα′ϕ]T (C(0))F∇T (e1, e2)

)e1 ∧ e2 .

58

It follows that

ddtS

(C(0),C(2),B)

CS/WZ (ϕT ,∇T )C

= T1

∫U

Tr ∂t( n∑i,j=1

ϕ]T (Cij)DTe1ϕ

]T (yi)DT

e2ϕ]T (yj) + 2πα′ϕ]T (C(0))F∇T (e1, e2)

)e1 ∧ e2

= T1

∫∂U

iξT(IV,∂tϕT )

(e1 ∧ e2) + 2πα′T1

∫∂U

iξT(IV,∂t∇T )

(e1 ∧ e2)

+ T1

∫U

Tr

( n∑i,j=1

∂tϕ](yi)

(DTe2ϕ

]T (yj)ϕ]T (Cij)e

1 − DTe1ϕ

]T (yj)ϕ]T (Cij)e

2)

(∇he1e1 +∇he2e2)

)e1 ∧ e2

− T1

∫ ∫U

Tr

(∑i,j

∂tϕ]T (yi)

(DTe1

(DTe2ϕ

]T (yj) · ϕ]T (Cij)

)− DT

e2

(DTe1ϕ

]T (yj) · ϕ]T (Cij)

)))e1 ∧ e2

+ T1

∫U

Tr(∑i,j

∂tϕ]T (Cij)D

Te1ϕ

]T (yi)DT

e2ϕ]T (yj)

)e1 ∧ e2

+ 2πα′ T1

∫U

Tr(∂tϕ

]T (C(0)) · F∇T (e1, e2)

)e1 ∧ e2

+ 2πα′ T1

∫U

Tr

(ϕ]T (C(0))

(((∂t∇T )(e2) e1 − (∂t∇T )(e1) e2)(∇he1e1 +∇he2e2) − (∂t∇T )([e1, e2])

))e1 ∧ e2

− 2πα′ T1

∫U

Tr(DTe1ϕ

]T (C(0)) · (∂t∇T )(e2) − DT

e2ϕ]T (C(0)) · (∂t∇T )(e1)

)e1 ∧ e2 .

Here,

ξT(IV,∂tϕT ) := Tr(∑

i,j∂tϕ]T (yi)DT

e2ϕ]T (yj)ϕ]T (Cij)

)e1 − Tr

(∑i,j∂tϕ

]T (yi)DT

e1ϕ]T (yj)ϕ]T (Cij)

)e2 ,

ξT(IV,∂t∇T ) := Tr(ϕ]T (C(0)) · (∂t∇T )(e2)

)e1 − Tr

(ϕ]T (C(0)) · (∂t∇T )(e1)

)e2

in T∗(UT /T )C, with the first OCU -linear in ∂tϕT and the second OC

U -linear in ∂t∇T .

6.3.2.4 D-membrane world-volume (m = 3)

Denote

C(3) := C(3) + C(1) ∧B

=∑i,j,k

(Cijk + CiBjk + CjBki + CkBij) dyi ⊗ dyj ⊗ dyk =

∑i,j,k

Cijkdyi ⊗ dyj ⊗ dyk

in a local coordinate (y1, · · · , yn) of Y . For D-membrane world-volume, dimX = 3. Let(e1, e2, e3) be an orthonormal frame on an open set U ⊂ X; (e1, e2, e3) its dual co-frame. Then,over U ,

S(C(1),C(3),B)CS/WZ (ϕT ,∇T )C

= T2

∫U

Tr( n∑i,j,k=1

ϕ](Cijk)DTe1ϕ

]T (yi)DT

e2ϕ]T (yj)DT

e3ϕ]T (yk)

+ 2πα′∑

(λµν)∈Sym3

n∑i=1

(−1)(λµν)(ϕ]T (Ci)D

Tλϕ

]T (yi)F∇T (eµ, eν)

))e1 ∧ e2 ∧ e3

It follows that

59

ddtS

(C(1),C(3),B)

CS/WZ (ϕT ,∇T )C

= T2

∫U

Tr ∂t( n∑i,j,k=1

ϕ](Cijk)DTe1ϕ

]T (yi)DT

e2ϕ]T (yj)DT

e3ϕ]T (yk)

+ 2πα′∑

(λµν)∈Sym3

n∑i=1

(−1)(λµν)(ϕ]T (Ci)DTeλϕ

]T (yi)F∇T (eµ, eν)

))e1 ∧ e2 ∧ e3

= T2

∫∂U

iξT(IV,∂tϕT ;C(3))

(e1 ∧ e2 ∧ e3) + 4πα′ T2

∫∂U

iξT(IV,∂tϕT ;C(1))

(e1 ∧ e2 ∧ e3)

+ 2πα′ T2

∫∂U

iξT(IV,∂t∇T)

(e1 ∧ e2 ∧ e3)

+ T2

∫U

((Tr∑i,j,k

ϕ]T (Cijk)∂tϕ]T (yi)DT

e2ϕ]T (yj)DT

e3ϕ]T (yk)

)e1

−(

Tr∑i,j,k

ϕ]T (Cijk)∂tϕ]T (yi)DT

e1ϕ]T (yj)DT

e3ϕ]T (yk)

)e2

−(

Tr∑i,j,k

ϕ]T (Cijk)∂tϕ]T (yi)DT

e2ϕ]T (yj)DT

e1ϕ]T (yk)

)e3

)(∑3µ=1∇

heµeµ) e1 ∧ e2 ∧ e3

− T2

∫U

Tr∑i,j,k

∂tϕ]T (yi)

(DTe1

(ϕ]T (Cijk)DT

e2ϕ]T (yj)DT

e3ϕ]T (yk)

)− DT

e2

(ϕ]T (Cijk)DT

e1ϕ]T (yj)DT

e3ϕ]T (yk)

)−DT

e3

(ϕ]T (Cijk)DT

e2ϕ]T (yj)DT

e1ϕ]T (yk)

))e1 ∧ e2 ∧ e3

+ T2

∫U

Tr∑i,j,k

∂tϕ]T (Cijk)DT

e1ϕ]T (yi)DT

e2ϕ]T (yj)DT

e3ϕ]T (yk) e1 ∧ e2 ∧ e3

+ 4πα′ T2

∫U

((Tr∑i

ϕ]T (Ci)∂tϕ]T (yi)F∇T (e2, e3)

)e1

−(

Tr∑i

ϕ]T (Ci)∂tϕ]T (yi)F∇T (e1, e3)

)e2

+(

Tr∑i

ϕ]T (Ci)∂tϕ]T (yi)F∇T (e1, e2)

)e3

)(∑3µ=1∇

heµeµ) e1 ∧ e2 ∧ e3

− 4πα′ T2

∫U

Tr∑i

∂tϕ]T (yi)

(DTe1

(ϕ]T (Ci)F∇T (e2, e3)

)− DT

e2

(ϕ]T (Ci)F∇T (e1, e3)

)+DT

e3

(ϕ]T (Ci)F∇T (e1, e2)

))e1 ∧ e2 ∧ e3

+ 2πα′ T2

∫U

Tr∑

(λµν)∈Sym3

∑i

(−1)(λµν)∂tϕ]T (Ci)D

Teλϕ

]T (yi)F∇T (eµ, eν) e1 ∧ e2 ∧ e3

+ 2πα′ T2

∫U

((Tr∑i

ϕ]T (Ci)(DTe3ϕ

]T (yi)(∂t∇T )(e2) − DT

e2ϕ]T (yi)(∂t∇T )(e3)

))e1

+(

Tr∑i

ϕ]T (Ci)(DTe3ϕ

]T (yi)(∂t∇T )(e1) + DT

e1ϕ]T (yi)(∂t∇T )(e3)

))e2

+(

Tr∑i

ϕ]T (Ci)(DTe1ϕ

]T (yi)(∂t∇T )(e2) − DT

e2ϕ]T (yi)(∂t∇T )(e1)

))e3

)(∑3µ=1∇

heµeµ)

e1 ∧ e2 ∧ e3

+ 2πα′ T2

∫U

Tr∑

(λµν)∈Sym3

∑i

(−1)(λµν)(ϕ]T (Ci)

[(∂t∇T )(eλ), ϕ]T (yi)

]F∇T (eµ, eν)

−ϕ]T (Ci)DTeλϕ

]T (yi) (∂tϕT )([eµ, eν ])

)e1 ∧ e2 ∧ e3 .

60

Here,

ξT(IV,∂tϕT ;C(3)) :=(

Tr∑i,j,k

ϕ]T (Cijk)∂tϕ]T (yi)DT

e2ϕ]T (yj)DT

e3ϕ]T (yk)

)e1

−(

Tr∑i,j,k

ϕ]T (Cijk)∂tϕ]T (yi)DT

e1ϕ]T (yj)DT

e3ϕ]T (yk)

)e2

−(

Tr∑i,j,k

ϕ]T (Cijk)∂tϕ]T (yi)DT

e2ϕ]T (yj)DT

e1ϕ]T (yk)

)e3 ,

ξT(IV,∂tϕT ;C(1)) :=(

Tr∑i

ϕ]T (Ci)∂tϕ]T (yi)F∇T (e2, e3)

)e1

−(

Tr∑i

ϕ]T (Ci)∂tϕ]T (yi)F∇T (e1, e3)

)e2 +

(Tr∑i

ϕ]T (Ci)∂tϕ]T (yi)F∇T (e1, e2)

)e3 ,

ξT(IV,∂t∇T ) :=(

Tr∑i

ϕ]T (Ci)(DTe3ϕ

]T (yi)(∂t∇T )(e2) − DT

e2ϕ]T (yi)(∂t∇T )(e3)

))e1

+(

Tr∑i

ϕ]T (Ci)(DTe3ϕ

]T (yi)(∂t∇T )(e1) + DT

e1ϕ]T (yi)(∂t∇T )(e3)

))e2

+(

Tr∑i

ϕ]T (Ci)(DTe1ϕ

]T (yi)(∂t∇T )(e2) − DT

e2ϕ]T (yi)(∂t∇T )(e1)

))e3

in T∗(UT /T )C, with the first two OCU -linear in ∂tϕT and the third OC

U -linear in ∂t∇T .

7 The second variation of the enhanced kinetic term for maps

Let T = (−ε, ε)2 ⊂ R2, with coordinate (s, t), and (ϕT ,∇T ) be an (∗2)-admissible family of(∗1)-admissible pairs, with (ϕ(0,0),∇(0,0)) = (ϕ,∇). Assume further that

Dξ∂sAϕT ⊂ Comm (AϕT ) for all ξ ∈ T∗(XT /T ) .

We work out in this section the second variation formula of the enhanced kinetic termS

(ρ,h;Φ,g)

map:kinetic+(ϕ,∇) in the standard action S(ρ,h;Φ,g,B,C)standard (ϕ,∇).

7.1 The second variation of the kinetic term for maps

Recall

E∇T(ϕT ) := S

(h;g)map:kinetic(ϕT ,∇

T ) :=1

2Tm−1

∫X

Tr⟨DTϕT , D

TϕT⟩

(h,g)vol h ,

with the understanding that all expressions are taken on X(s,t) with (s, t) varying in T .Let U ⊂ X be an open set with an orthonormal frame (eµ)µ=1, ··· ,m. Let (eµ)µ=1, ··· ,m be the

dual co-frame. Assume that U is small enough so that ϕT (UAzT ) is contained in a coordinatechart of Y , with coordinates (y1, · · · , yn). Then, as in Sec. 6.1, over U ,

∂

∂tE∇

T

(ϕT ) = Tm−1

∫U

Tr

m∑µ=1

⟨∇T,(ϕT ,g)eµ ∂tϕT , D

TeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr

m∑µ=1

⟨(ad ⊗∇g)DTeµϕT ∂tϕT , D

TeµϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi , DTeµϕT

⟩g

vol h

=: (I 2.1) + (I 2.2) + (I 2.3) ;

61

and∂2

∂s ∂tE∇

T(ϕT ) =

∂

∂s(I 2.1) +

∂

∂s(I 2.2) +

∂

∂s(I 2.3) .

Which we now compute term by term.

The term ∂∂s (I 2.1)

∂

∂s(I 2.1) = Tm−1

∂

∂s

∫U

Trm∑µ=1

⟨∇T,(ϕT ,g)eµ ∂tϕT , D

TeµϕT

⟩g

vol h

= Tm−1

∫U

Tr∑µ

∂s

⟨∇T,(ϕT ,g)eµ ∂tϕT , D

TeµϕT

⟩g

vol h

= Tm−1

∫U

Tr∑µ

⟨∂s∇T,(ϕT ,g)eµ ∂tϕT , D

TeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨∇T,(ϕT ,g)eµ ∂tϕT , ∂sD

TeµϕT

⟩g

vol h

= (I 2.1.1) + (I 2.1.2) .

(a) Term (I 2.1.1)

(I 2.1.1) := Tm−1

∫U

Tr∑µ

⟨∂s∇T,(ϕT ,g)eµ ∂tϕT , D

TeµϕT

⟩g

vol h

= Tm−1

∫U

Tr∑µ

⟨∇T,(ϕT ,g)eµ ∂s∂tϕT , D

TeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨F∇T,(ϕT ,g)(∂s, eµ) ∂tϕT , D

TeµϕT

⟩g

vol h

= (I 2.1.1.1) + (I 2.1.1.2) .

For Term (I 2.1.1.1), as in Sec. 6.1 for Summand (I.1.1), consider the 1-form on UT /T

αT(I2,∂s∂tϕT ) := Tr⟨∂s∂tϕT , D

TϕT⟩g

and let

ξT(I2,∂s∂tϕT ) :=

n∑µ=1

Tr⟨∂s∂tϕT , D

TeµϕT

⟩geµ

be its dual on UT /T with respect to h. Then,

(I 2.1.1.1) = Tm−1

∫∂UiξT

(I2,∂s∂tϕT )vol h

+ Tm−1

∫U

Tr⟨∂s∂tϕT ,

(DT∑

µ∇heµeµ−∑

µ∇T,(ϕT ,g)eµ DT

eµ

)ϕT

⟩g

vol h .

For Term (I 2.1.1.2), recall Lemma 3.2.2.5. Then,

(I 2.1.1.2) = − Tm−1

∫U

Tr⟨∂tϕT ,

∑µ

F∇T,(ϕT ,g)(∂s, eµ)DTeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

[F∇(∂s, eµ) , 〈∂tϕT , DT

eµϕT 〉g]

vol h

= − Tm−1

∫U

Tr⟨∂tϕT ,

∑µ

F∇T,(ϕT ,g)(∂s, eµ)DTeµϕT

⟩g

vol h .

62

Here,

F∇T,(ϕT ,g)(∂s, eµ)DTeµϕT =

(∂s∇T,(ϕT ,g)eµ − ∇T,(ϕT ,g)eµ ∂s

)∑n

i=1Deµϕ

]T (yi)⊗ ∂

∂yi

=

n∑i=1

[(∂s∇T )(eµ), Deµϕ]T (yi)]⊗ ∂

∂yi+

n∑i=1

Deµϕ]T (yi)

n∑j=1

[(∂s∇T )(eµ), ϕ]T (yj)]⊗∇g∂∂yj

∂∂yi

+

n∑i=1

Deµϕ]T (yi)

n∑j,k=1

(DTeµϕ

]T (yj) ∂sϕ

]T (yk)⊗∇g ∂

∂yk

∇g∂∂yj

∂∂yi

− ∂sϕ]T (yk)DTeµϕ

]T (yj)⊗∇g∂

∂yj

∇g ∂∂yk

∂∂yi

)explicitly.

(b) Term (I 2.1.2)

(I 2.1.2) := Tm−1

∫U

Tr∑µ

⟨∇T,(ϕT ,g)eµ ∂tϕT , ∂sD

TeµϕT

⟩g

vol h

= Tm−1

∫U

Tr∑µ

⟨∇T,(ϕT ,g)eµ ∂tϕT , ∇T,(ϕT ,g)eµ ∂sϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨∇T,(ϕT ,g)eµ ∂tϕT , (ad ⊗∇g)DTeµϕT ∂sϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨∇T,(ϕT ,g)eµ ∂tϕT ,

∑ni=1

[(∂s∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi

⟩g

vol h .

As in Sec. 6.1, consider the 1-forms on UT /T ,

αT(I2,∂tϕT ,∇T,(ϕT ,g))

= Tr⟨∂tϕT , ∇T,(ϕT ,g)∂sϕT

⟩g,

αT(I2,∂tϕT ,DTϕT ) = Tr⟨∂tϕT , (ad ⊗∇g)DTϕT ∂sϕT

⟩g,

αT(I2,∂tϕT ,∂s∇T ) = Tr⟨∂tϕT ,

∑ni=1[∂s∇T , ϕ]T (yi)]⊗ ∂

∂yi

⟩g

and let

ξT(I2,∂tϕT ,∇T,(ϕT ,g))

=∑µ

Tr⟨∂tϕT , ∇T,(ϕT ,g)eµ ∂sϕT

⟩geµ ,

ξT(I2,∂tϕT ,DTϕT ) =∑µ

Tr⟨∂tϕT , (ad ⊗∇g)DTeµϕT ∂sϕT

⟩geµ ,

ξT(I2,∂tϕT ,∂s∇T ) =∑µ

Tr⟨∂tϕT ,

∑ni=1[(∂s∇T )(eµ), ϕ]T (yi)]⊗ ∂

∂yi

⟩geµ

be their respective dual on UT /T with respect to h. Then,

(I 2.1.2) = Tm−1

∫∂U

i ξT(I2,∂tϕT ,∇

T,(ϕT ,g))+ ξT

(I2,∂tϕT ,DT ϕT )

+ ξT(I2,∂tϕT ,∂s∇T )

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

(∇T,(ϕT ,g)∑

µ∇heµeµ−∑

µ∇T,(ϕT ,g)eµ ∇T,(ϕT ,g)eµ

)∂sϕT

⟩g

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

((ad ⊗∇g)DT∑

µ∇heµeµϕT

−∑

µ∇T,(ϕT ,g)eµ (ad ⊗∇g)DTeµϕT

)∂sϕT

⟩g

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

∑ni=1

([(∂s∇T )(

∑µ∇

heµeµ), ϕ]T (yi)

]−∑

µ∇T,(ϕT ,g)eµ

[(∂s∇T )(eµ), ϕ]T (yi)

])⊗ ∂

∂yi

⟩g

vol h .

63

The term ∂∂s (I 2.2)

∂

∂s(I 2.2) = Tm−1

∂

∂s

∫U

Tr

m∑µ=1

⟨(ad ⊗∇g)DTeµϕT ∂tϕT , D

TeµϕT

⟩g

vol h

= Tm−1

∫U

Tr∑µ

⟨∂s((ad ⊗∇g)DeµϕT ∂tϕT

), DT

eµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DeµϕT ∂tϕT , ∂sD

TeµϕT

⟩g

vol h

= (I 2.2.1) + (I 2.2.2) .

(a) Term (I 2.2.1)

(I 2.2.1) = Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DTeµ (∂s∂tϕT ) , DT

eµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨∑i,j

[DTeµ∂sϕ

]T (yj), ∂tϕ

]T (yi)

]⊗∇g∂

∂yj

∂∂yi , D

TeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨∑i,j,k

([DTeµϕ

]T (yj), ∂tϕ

]T (yi)

]∂sϕ

]T (yk)⊗Rg

( ∂∂yk

, ∂∂yj) ∂∂yi

− ∂tϕ]T (yi)[DTeµϕ

]T (yj), ∂sϕ

]T (yk)

]⊗∇g∂

∂yj

∇g ∂∂yk

∂∂yi

),

DTeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨∑i,j

[ad (∂s∇T )(eµ)ϕ

]T (yj), ∂tϕ

]T (yi)

]⊗∇g∂

∂yj

∂∂yi

, DTeµϕT

⟩g

vol h .

The integrand of the first summand captures a related part in the system of equations of motionfor (ϕ,∇). The integrand of the second summand is tensorial in ∂tϕT and first-order differentialoperatorial in ∂sϕT . The integrand of the third summand is tensorial in both ∂tϕT and ∂sϕT .The integrand of the fourth summand is tensorial in ∂tϕT and ∂s∇T .

(b) Term (I 2.2.2)

(I 2.2.2) = Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DeµϕT ∂tϕT , ∂sD

TeµϕT

⟩g

vol h

= Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DeµϕT ∂tϕT , ∇

T,(ϕT ,g)eµ ∂sϕT

⟩g

vol h

− Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DeµϕT ∂tϕT , (ad ⊗∇g)∂sϕTDT

eµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DeµϕT ∂tϕT ,

∑i

[(∂s∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi

⟩g

vol h .

The integrand of the first summand is tensorial in ∂tϕT and first-order differential operatorial in∂sϕT . The integrand of the second summand is tensorial in both ∂tϕT and ∂sϕT . The integrandof the third summand is tensorial in ∂tϕT and ∂s∇T .

The term ∂∂s (I 2.3)

64

∂

∂s(I 2.3) = Tm−1

∂

∂s

∫U

m∑µ=1

⟨ n∑i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, DT

eµϕT

⟩g

vol h

= Tm−1

∫U

m∑µ=1

⟨ n∑i=1

∂s

([(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi

), DT

eµϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, ∂sD

TeµϕT

⟩g

vol h

= (I 2.3.1) + (I 2.3.2) .

(a) Term (I 2.3.1)

(I 2.3.1) = Tm−1

∫U

m∑µ=1

⟨ n∑i=1

∂s

([(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi

), DT

eµϕT

⟩g

vol h

= Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂s∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, DT

eµϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

([(∂t∇T )(eµ), ∂sϕ

]T (yi)

]⊗ ∂

∂yi

+[(∂t∇T )(eµ), ϕ]T (yi)

]∑j∂sϕ

]T (yj)⊗∇g∂

∂yj

∂∂yi

), DT

eµϕT

⟩g

vol h .

The integrand of the first summand captures a related part in the system of equations of motionfor (ϕ,∇). The integrand of the second summand is tensorial in ∂sϕT and ∂t∇T .

(b) Term (I 2.3.2)

(I 2.3.2) = Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, ∂sD

TeµϕT

⟩g

vol h

= Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, ∇T,(ϕT ,g)eµ ∂sϕT

⟩g

vol h

− Tm−1

∫U

m∑µ=1

⟨∑n

i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, (ad ⊗∇g)∂sϕTDT

eµϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

n∑i,j=1

⟨[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi,[(∂s∇T )(eµ), ϕ]T (yj)

]⊗ ∂

∂yj

⟩g

vol h .

The integrand of the first summand is tensorial in ∂t∇T and first-order differential operatorialin ∂sϕT . The integrand of the second summand is tensorial in ∂sϕT and ∂t∇T . The integrandof the third summand is tensorial in both ∂t∇T and ∂s∇T .

Finally, recall Lemma 3.2.2.4 and note that with the additional assumption at the beginningof this section, all the inner products Tr 〈 · , · 〉g that appear in the calculation above are defined.

In summary,

Proposition 7.1.1. [second variation of kinetic term for maps] Let (ϕT ,∇T ) be a (∗2)-admissible T -family of (∗1)-admissible pairs with the additional assumption that

65

Dξ∂sAϕT ⊂ Comm (AϕT ) for all ξ ∈ T∗(XT /T ). Then,

∂∂s

∂∂tE∇

T

(ϕT )C =∂∂s

∂∂t

(12Tm−1

∫U

Tr 〈DTϕT , DTϕT 〉(h,g) vol h

)= Tm−1

∫∂U

iξT(I2,∂s∂tϕT )

vol h

+ Tm−1

∫∂U

i ξT(I2,∂tϕT ,∇

T,(ϕT ,g))+ ξT

(I2,∂tϕT ,DT ϕT )

+ ξT(I2,∂tϕT ,∂s∇T )

vol h

+ Tm−1

∫U

Tr⟨∂s∂tϕT ,

(DT∑

µ∇heµeµ−∑µ∇

T,(ϕT ,g)eµ DT

eµ

)ϕT⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DTeµ (∂s∂tϕT ) , DT

eµϕT⟩g

vol h

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂s∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂∂yi

, DTeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

(∇T,(ϕT ,g)∑

µ∇heµeµ−∑

µ∇T,(ϕT ,g)eµ ∇T,(ϕT ,g)eµ

)∂sϕT

⟩g

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

((ad ⊗∇g)DT∑

µ∇heµeµϕT

−∑

µ∇T,(ϕT ,g)eµ (ad ⊗∇g)DTeµϕT

)∂sϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨∑i,j

[DTeµ∂sϕ

]T (yj), ∂tϕ

]T (yi)

]⊗∇g ∂

∂yj

∂∂yi , D

TeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨∑i,j,k

([DTeµϕ

]T (yj), ∂tϕ

]T (yi)

]∂sϕ

]T (yk)⊗Rg

( ∂∂yk

,∂∂yj) ∂∂yi

− ∂tϕ]T (yi)[DTeµϕ

]T (yj), ∂sϕ

]T (yk)

]⊗∇g ∂

∂yj

∇g ∂∂yk

∂∂yi

),

DTeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DeµϕT ∂tϕT , ∇

T,(ϕT ,g)eµ ∂sϕT

⟩g

vol h

− Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DeµϕT ∂tϕT , (ad ⊗∇g)∂sϕTD

TeµϕT

⟩g

vol h

− Tm−1

∫U

Tr⟨∂tϕT ,

∑µ

F∇T,(ϕT ,g)(∂s, eµ)DTeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

∑ni=1

([(∂s∇T )(

∑µ∇

heµeµ), ϕ]T (yi)

]−∑

µ∇T,(ϕT ,g)eµ

[(∂s∇T )(eµ), ϕ]T (yi)

])⊗ ∂∂yi

⟩g

vol h .

+ Tm−1

∫U

Tr∑µ

⟨∑i,j

[ad (∂s∇T )(eµ)ϕ

]T (yj), ∂tϕ

]T (yi)

]⊗∇g ∂

∂yj

∂∂yi

, DTeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DeµϕT ∂tϕT ,

∑i

[(∂s∇T )(eµ), ϕ]T (yi)

]⊗ ∂∂yi

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

([(∂t∇T )(eµ), ∂sϕ

]T (yi)

]⊗ ∂∂yi

+[(∂t∇T )(eµ), ϕ]T (yi)

]∑j∂sϕ

]T (yj)⊗∇g ∂

∂yj

∂∂yi

), DT

eµϕT⟩g

vol h

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂∂yi

, ∇T,(ϕT ,g)eµ ∂sϕT⟩g

vol h

− Tm−1

∫U

m∑µ=1

⟨∑n

i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂∂yi

, (ad ⊗∇g)∂sϕTDTeµϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

n∑i,j=1

⟨[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂∂yi

,[(∂s∇T )(eµ), ϕ]T (yj)

]⊗ ∂∂yj

⟩g

vol h .

66

Here,

ξT(I2,∂s∂tϕT ) :=

n∑µ=1

Tr⟨∂s∂tϕT , D

TeµϕT

⟩geµ ,

ξT(I2,∂tϕT ,∇T,(ϕT ,g))

=∑µ

Tr⟨∂tϕT , ∇T,(ϕT ,g)eµ ∂sϕT

⟩geµ ,

ξT(I2,∂tϕT ,DTϕT ) =∑µ

Tr⟨∂tϕT , (ad ⊗∇g)DTeµϕT ∂sϕT

⟩geµ ,

ξT(I2,∂tϕT ,∂s∇T ) =∑µ

Tr⟨∂tϕT ,

∑ni=1[(∂s∇T )(eµ), ϕ]T (yi)]⊗ ∂

∂yi⟩geµ

and

F∇T,(ϕT ,g)(∂s, eµ)DTeµϕT =

(∂s∇T,(ϕT ,g)eµ − ∇T,(ϕT ,g)eµ ∂s

)∑n

i=1Deµϕ

]T (yi)⊗ ∂

∂yi

=

n∑i=1

[(∂s∇T )(eµ), Deµϕ]T (yi)]⊗ ∂

∂yi+

n∑i=1

Deµϕ]T (yi)

n∑j=1

[(∂s∇T )(eµ), ϕ]T (yj)]⊗∇g∂∂yj

∂∂yi

+n∑i=1

Deµϕ]T (yi)

n∑j,k=1

(DTeµϕ

]T (yj) ∂sϕ

]T (yk)⊗∇g ∂

∂yk

∇g∂∂yj

∂∂yi

− ∂sϕ]T (yk)DTeµϕ

]T (yj)⊗∇g∂

∂yj

∇g ∂∂yk

∂∂yi

)The summands

+ Tm−1

∫U

Tr⟨∂s∂tϕT ,

(DT∑

µ∇heµeµ−∑

µ∇T,(ϕT ,g)eµ DT

eµ

)ϕT

⟩g

vol h

+ Tm−1

∫U

Tr∑µ

⟨(ad ⊗∇g)DTeµ (∂s∂tϕT ) , DT

eµϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂s∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, DT

eµϕT

⟩g

vol h

will vanish when imposing the equations of motion for (ϕ,∇).

67

If (ϕT ,∇T ) is furthermore a (∗2)-admissible T -family of (∗2)-admissible pairs, Then, theabove expression reduces to

∂∂s

∂∂tE∇

T

(ϕT )C = ∂∂s

∂∂t

(12Tm−1

∫U

Tr 〈DTϕT , DTϕT 〉(h,g) vol h

)= Tm−1

∫∂U

iξT(I2,∂s∂tϕT )

vol h

+ Tm−1

∫∂U

i ξT(I2,∂tϕT ,∇

T,(ϕT ,g))+ ξT

(I2,∂tϕT ,DT ϕT )

+ ξT(I2,∂tϕT ,∂s∇T )

vol h

+ Tm−1

∫U

Tr⟨∂s∂tϕT ,

(DT∑

µ∇heµeµ−∑µ∇

T,(ϕT ,g)eµ DT

eµ

)ϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂s∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, DT

eµϕT

⟩g

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

(∇T,(ϕT ,g)∑

µ∇heµeµ−∑

µ∇T,(ϕT ,g)eµ ∇T,(ϕT ,g)eµ

)∂sϕT

⟩g

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

((ad ⊗∇g)DT∑

µ∇heµeµϕT

−∑

µ∇T,(ϕT ,g)eµ (ad ⊗∇g)DTeµϕT

)∂sϕT

⟩g

vol h

− Tm−1

∫U

Tr⟨∂tϕT ,

∑µ

F∇T,(ϕT ,g)(∂s, eµ)DTeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

∑ni=1

([(∂s∇T )(

∑µ∇

heµeµ), ϕ]T (yi)

]−∑

µ∇T,(ϕT ,g)eµ

[(∂s∇T )(eµ), ϕ]T (yi)

])⊗ ∂

∂yi

⟩g

vol h .

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

([(∂t∇T )(eµ), ∂sϕ

]T (yi)

]⊗ ∂

∂yi

+[(∂t∇T )(eµ), ϕ]T (yi)

]∑j∂sϕ

]T (yj)⊗∇g∂

∂yj

∂∂yi

), DT

eµϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, ∇T,(ϕT ,g)eµ ∂sϕT

⟩g

vol h

− Tm−1

∫U

m∑µ=1

⟨∑n

i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, (ad ⊗∇g)∂sϕTDT

eµϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

n∑i,j=1

⟨[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi,[(∂s∇T )(eµ), ϕ]T (yj)

]⊗ ∂

∂yj

⟩g

vol h .

68

If one further imposes the equations of motion on (ϕ,∇), then the expression reduces furtherto

∂∂s

∂∂tE∇

T

(ϕT )C = ∂∂s

∂∂t

(12Tm−1

∫U

Tr 〈DTϕT , DTϕT 〉(h,g) vol h

)= Tm−1

∫∂U

iξT(I2,∂s∂tϕT )

vol h

+ Tm−1

∫∂U

i ξT(I2,∂tϕT ,∇

T,(ϕT ,g))+ ξT

(I2,∂tϕT ,DT ϕT )

+ ξT(I2,∂tϕT ,∂s∇T )

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

(∇T,(ϕT ,g)∑

µ∇heµeµ−∑

µ∇T,(ϕT ,g)eµ ∇T,(ϕT ,g)eµ

)∂sϕT

⟩g

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

((ad ⊗∇g)DT∑

µ∇heµeµϕT

−∑

µ∇T,(ϕT ,g)eµ (ad ⊗∇g)DTeµϕT

)∂sϕT

⟩g

vol h

− Tm−1

∫U

Tr⟨∂tϕT ,

∑µ

F∇T,(ϕT ,g)(∂s, eµ)DTeµϕT

⟩g

vol h

+ Tm−1

∫U

Tr⟨∂tϕT ,

∑ni=1

([(∂s∇T )(

∑µ∇

heµeµ), ϕ]T (yi)

]−∑

µ∇T,(ϕT ,g)eµ

[(∂s∇T )(eµ), ϕ]T (yi)

])⊗ ∂

∂yi

⟩g

vol h .

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

([(∂t∇T )(eµ), ∂sϕ

]T (yi)

]⊗ ∂

∂yi

+[(∂t∇T )(eµ), ϕ]T (yi)

]∑j∂sϕ

]T (yj)⊗∇g∂

∂yj

∂∂yi

), DT

eµϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

⟨ n∑i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, ∇T,(ϕT ,g)eµ ∂sϕT

⟩g

vol h

− Tm−1

∫U

m∑µ=1

⟨∑n

i=1

[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi, (ad ⊗∇g)∂sϕTDT

eµϕT

⟩g

vol h

+ Tm−1

∫U

m∑µ=1

n∑i,j=1

⟨[(∂t∇T )(eµ), ϕ]T (yi)

]⊗ ∂

∂yi,[(∂s∇T )(eµ), ϕ]T (yj)

]⊗ ∂

∂yj

⟩g

vol h .

7.2 The second variation of the dilaton term

We now work out the second variation of the (complexified) dilaton term

S(ρ,h;Φ)dilaton (ϕT )C =

∫U

Tr 〈dρ , ϕTdΦ〉h vol h

=

∫U

Tr(∑

µ

dρ(eµ) ((DT

eµϕT )Φ))

vol h .

for an (∗1)-admissible family of (∗1)-admissible pairs (ϕT ,∇T ), T := (−ε, ε)2 ⊂ R2 with coordi-nate (s, t).

It follows from Sec. 6.2 that, due to the effect of the trace map Tr ,

∂

∂tS

(ρ,h;Φ)dilaton (ϕT )C =

∫U

Tr

m∑µ=1

dρ(eµ) ∂t((DT

eµϕT )Φ)vol h

=

∫U

Tr∑µ

dρ(eµ)DTeµ

((∂tϕT )Φ

)vol h .

69

Thus, due to the effect of the trace map Tr again,

∂

∂s

∂

∂tS

(ρ,h;Φ)dilaton (ϕT )C =

∫U

Tr∑µ

dρ(eµ)∂sDTeµ

((∂tϕT )Φ

)vol h

=

∫U

Tr∑µ

dρ(eµ)DTeµ∂s

((∂tϕT )Φ

)vol h

=

∫U

Tr∑µ

dρ(eµ)DTeµ

((∂s∂tϕT )Φ

)vol h

+

∫U

Tr∑µ

dρ(eµ)DTeµ

(∑i,j

∂tϕ]T (yi)∂sϕ

]T (yj)⊗

(∂∂yi

∂∂yj−∇g∂

∂yi

∂∂yj

)Φ)

vol h

= (II 2.1) + (II 2.2) .

For Summand (II 2.1), repeating the same argument in Sec. 6.1 for Summand (I.1.1), oneconcludes that

(II 2.1) =

∫∂UiξT

(II2,∂s∂tϕT )vol h

+

∫U

(∑µ

∇heµeµ −∑µ

eµdρ(eµ))

Tr((∂s∂tϕT )Φ

)vol h ,

whereξT

(II2,∂s∂tϕT ):=

∑µ

(dρ(eµ)Tr

((∂s∂tϕT )Φ

))eµ ∈ T∗(UT /T ) .

The second summand of Summand (II 2.1) above is the term that captures the S(ρ,h;Φ)dilaton (ϕ)-

contribution to the system of equations of motion for (ϕ,∇).

With ∂s∂tϕT replaced by∑

i,j∂tϕ]T (yi)∂sϕ

]T (yj)⊗

(∂∂yi

∂∂yj−∇g∂

∂yi

∂∂yj

)Φ, one has similarly

(II 2.2) =

∫∂UiξT

(II2,∂tϕT ,∂sϕT )vol h

+

∫U

(∑µ

∇heµeµ −∑µ

eµdρ(eµ))·

Tr(∑

i,j

∂tϕ]T (yi)∂sϕ

]T (yj)⊗

(∂∂yi

∂∂yj−∇g∂

∂yi

∂∂yj

)Φ)

vol h ,

where

ξT(II2,∂tϕT ,∂sϕT )

:=∑µ

(dρ(eµ)Tr

(∑i,j

∂tϕ]T (yi)∂sϕ

]T (yj)⊗

(∂∂yi

∂∂yj−∇g∂

∂yi

∂∂yj

)Φ))eµ

in T∗(UT /T ). The second summand of Summand (II 2.2) above contributes to the zeroth order

terms of the differential operator on (∂sϕT , ∂tϕT ) from the second variation of S(ρ,h;Φ,g,B,C)standard (ϕ,∇).

In summary,

Proposition 7.2.1. [second variation of S(ρ,h;Φ)dilaton (ϕ)C] For the (complexified) dilaton term

S(ρ,h;Φ)dilaton (ϕ)C :=

∫U

Tr 〈dρ , ϕdΦ〉h vol h ,

70

its second variation for a (∗1)-admissible family of (∗1)-admissible pairs (ϕT ,∇T ), T := (−ε, ε)2 ⊂R2 with coordinate (s, t), is given by

∂

∂s

∂

∂tS

(ρ,h;Φ)dilaton (ϕT )C

=

∫∂U

iξT(II2,∂s∂tϕT )

+ξT(II2,∂tϕT ∂sϕT )

vol h

+

∫U

(∑µ

∇heµeµ −∑µ

eµdρ(eµ))

Tr((∂s∂tϕT )Φ

)vol h

+

∫U

(∑µ

∇heµeµ −∑µ

eµdρ(eµ))·

Tr(∑i,j

∂tϕ]T (yi)∂sϕ

]T (yj)⊗

(∂∂yi

∂∂yj−∇g∂

∂yi

∂∂yj

)Φ)

vol h ,

where

ξT(II2,∂s∂tϕT ) :=∑µ

(dρ(eµ)Tr

((∂s∂tϕT )Φ

))eµ

ξT(II2,∂tϕT ,∂sϕT ) :=∑µ

(dρ(eµ)Tr

(∑i,j

∂tϕ]T (yi)∂sϕ

]T (yj)⊗

(∂∂yi

∂∂yj−∇g∂

∂yi

∂∂yj

)Φ))eµ

in T∗(UT /T ). The integral∫U

(∑µ

∇heµeµ −∑µ

eµdρ(eµ))

Tr((∂s∂tϕT )Φ

)vol h

would vanish when imposing the equations of motion of (ϕ,∇) after the combination with other

Equations-of-Motion capturing parts from the second variation of other terms in S(ρ,h;Φ,g,B,C)standard (ϕ,∇)C.

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Where we are

The following table summarizes where we are, following the similar steps of fundamental strings

string theory D-brane theory

fundamental objects :open or closed string ,

fundamental objects :Azumaya/matrix

(m− 1)-manifold

with a fundamental module

with a connection

string world-sheet :2-manifold Σ

D-brane world-volume :Azumaya/matrix m-manifoldwith a fundamental module with a connection

(XAz, E ;∇)

string moving in space-time Y :differentiable map f : Σ→ Y

D-brane moving in space-time Y :(admissible) differentiable map ϕ : (XAz, E ;∇)→ Y

Nambu-Goto action SNG for f ’s Dirac-Born-Infeld action SDBI for (ϕ,∇)’s

Polyakov action SPolyakov for f ’s standard action Sstandard for (ϕ,∇)’s

action for Ramond-Neveu-Schwarz superstrings ???, cf. [L-Y6: Sec. 5.1] (D(11.2))

action for Green-Schwarz superstrings ???, cf. [L-Y6: Sec. 5.1] (D(11.2))

quantization ???

(Cf. [L-Y8: Remark 3.2.4: second table]) (D(13.1)). It’s by now a history that as the built-in structure ofa string is far richer than that for a point, a physical theory that takes strings as fundamental objects hasbrought us to where a physical theory that takes only point-particles as fundamental objects cannot reach.Now that a D-brane carries even more built-in structures, are these even-richer-than-string structures alljust in vain? Or is a physical theory that takes D-branes as fundamental objects going to lead us tosomewhere beyond that from string theories?

Besides a theory in its own right, a theory that takes D-branes as fundamental objects has deepconnection with other themes outside. In particular, at low dimensions, that there should be the followingconnections are “obvious”

(0) (m = 0) =⇒ a new class of matrix models; cf. [L-Y8: Figure 2-1-2] (D(13.1))(1) (m = 1) =⇒ nature of non-Abelian Ramond-Ramond fields; cf. e− vs. EM field, [Ja](2) (m = 2) =⇒ a new Gromov-Witten type theory; cf. [L-Y3] (D(10.1)), [L-Y4] (D(10.2))

but most details to realize these connections remain far from reach at the moment.

• A reflection at the end of the first decade of the D-project since spring 2007:

我到為種植，我行花未開，豈無佳色在，留待後人來。 ～～～ 弘一法師 （李叔同,1880-1942）: 『將離淨峰詠菊誌別』

I came to plant seedsWhen I departed, the flowers hadn’t yet blossomedNot that there is no beautiful sceneOnly left to new generations ～～～ Master Hong Yi (1880-1942): “A zen poem on chrysanthemum” (English translation by Ling-Miao Chou)

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chienhao.liu@gmail.com, chienliu@cmsa.fas.harvard.edu;yau@math.harvard.edu

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